VP 3 



vOO. 



c^ -V, 



•^ «>' 



/. -> 






x>\^^ 



.0^ 






^.<^^- 




.^^ 






-^^^ 



v' 



x^^' ^/ 









\ 



^. 



..\^' -^^ '■ . -'^ 



■^^ c ° '^ '•^ 



• 0- 



^^"^ J^ 



.J> \^ 






x^'^> 



'"■>>" .,^'^f^. '- 






-^^ 



o. .0- 









.^^ 



</• \ 






A^- '^^, 



,# 
















^^x:"^ .'.^?5^,-. 






.^' ,<h^ 



^J^^. 









,-K^ 



lA- V 









^^^ ^^. ^. 



o.^;- 












,s\ 



■^.•^_ 1. ^1 



%^- 



<.^^^- 









^-^ ,A^' 



.N^ -<^. 



:^ 



.. ^.>i^t^ 



vXSS3j> >" 


















.\^ 






.0 5. 



a\ , N <- ^ -/' - ^ ^-v 



%-4 






•is' 






^S^"' 

(^^' -^o. 



























■ O' 



^OQ^ 



o>' 



-«^" , _ --Sir 










PRODROMUS 



MATHEMATICAL AETS 



CONTAINING DIRECTIONS FOR 



SURVEYING AND ENGINEERING. 



BY AMOS EATON, A. B. & A. M., 

'oniur Professor iii Kensselaer Institute, and Prof. Civil Engineering. Ten years an acting 

1/and Agent, Surveyor, and Engineer ; while pursuing the profession of Law. Member 

of the Amer. Geol. Soc— ofPhil. Acad. Nat. Sci— of N. York Lye. Nat. Hisi.,&c. 




TROY, N. Y. 



PUBLISHED BY ELIAS GATES. 

ALBANY, O. STEELE ; NEjW-YORKJ' ROBINSON, PRATT & CO. ; 
PHILADELPHIA, HENRY PERKINS. 

TOTTLE, BELCHER AND BURTON, PRINTERS. 

1838. 






If 



Entered according to Act of Congress, in tlie year J838, by iJio pioprietor, Elias Gates, in the 
Clerk's Office of tiie District Court of the Nortliern District of New- York. 



3 3 ^ t- 



^-J?74.6. 



F R E F A C K . 



This book is chiefly made up of selections from a mass of hete- 
rogeneous materials, which I have been depositing in my common 
journal for more than thirty years — some of which, howevei', 
1 published in 1830, under the title " Art without Science." I may 
add, that I had published a very small treatise under the same title, 
in the year 1800. 

Students have made use of manuscript copies from my notes, for 
a kind of guide in their course of exercises, for the last four years. 
Examiners, appointed by the Patron of this Institute, have followed 
them, mostly, for the same period. 

Though it is offered as the Prodromus* of a full treatise on Ma- 
thematical Arts ; I have progressed too far on the way to the iourne 
of three score and ten, to give any assurances. I have materials 
enough to complete the object ; but " there is a point by nature 
fixed, whence life must downward tend." 

Logarithms are not used in this book for purposes of calculation. 
It is a mistake to suppose that logarithms expedite calculations in 
trigonometry, in common applications of it. The tedious processes 
of multiplication and division, when we use natural sines, are over- 
balanced by the trouble of looking out logarithms and accommo- 
dating them to the various cases. This opinion is to be proved or 
disproved by trial alone. But in a long tedious process of several 
days labor, logarithms are generally useful. 

Algebraic expressions are not used ; because they are unneces- 
saiy, and very few are sufficiently versed in algebra to apply them 
to advantage. In truth, our fives are too short to devote much time 
to speculative mathematics. In past ages, when the science of 
nature was in its infancy, more time could be devoted to " mere 
tricks to stretch the human brain," than in this day of astonishing 
developements of nature's wonders. 

* Prodromos, Greek, fore-runner. 



Every teacher of experience knows, that the only successful mode 
of instruction is that which interests the student. Also that it is^ 
exceedingly ditTicult to excite interest by a blindfold course, whose 
object is not perceived. In learning land surveying, the student 
should always survey, under the mechanical direction of a teacher, 
before he studies the science of surveying. He should take latitude 
and longitude, and learn the use of the necessary instruments, before 
he devotes a day to the theory of lunar observations, &c. He will 
then perceive the object of his closet studies, and hear, understand- 
ingly and with dehght, the lectures of his teacher. Even the common 
proportions of a triangle should not be introduced to a student's 
mind, until the teacher has directed him in the measurement of the 
lines and angles in true-earnest application. It is scarcely more 
absurd to attempt to theorize a blacksmith's boy into horse-shoeing, 
than to attempt to make a practical mathematician without out-of- 
doors practice. 

But the extreme of absurdity is most emphatically exhibited by 
putting books into students' hands, written in a language which they 
cannot understand. If they must read a language which is new to 
them, they must have time to learn it. An honest teacher ought, in 
such cases, to make the students or their guardians, understand that 
they are not to study mathematics, until they have devoted a year 
to the study of a new language, called algebra. 

In this small work, an attempt is made to enable a common-sense 
farmer, mechanic, merchant, or other man of business, who is but 
an ordinary arithmetician, to become sufficiently qualified for the 
business concerns of life, as a practical mathematician. But he 
must be shewn the use of mstruments ; as it is an idle waste of time 
to attempt to learn their use from books. 



AMOS EATON. 



Rensselaek Institute, 
Troy, March, 1838. 



POCKET SCALE 

Of Natural Sines, Chord Line, and Equal PaHs. 

A six inch pocket ruler has always been considered as essential 
to a mechanic. To a mathematician, or even an ordinary traveller, 
dzc., such a measure should furnish the necessary scales of what 
we need most. A scale of equal parts of an inch in tenths, and a 
diagonal inch for hundredths, are the most important. A line of 
chords, of about a three inch sweep of sixty, is generally deemed 
next in importance. Geometrical trigonometry may be wrought by 
these two scales. But our situation is not always (nor even at one 
occasion in one hundred) such, that we can sit down at a table, and 
use the scale and dividers accurately. But we can make calcula- 
tions with natural sines, while riding in a stage, or sitting at the 
theatre, or a concert. But this operation demands a table of natu- 
ral sines. A book, then, of many pages, must be carried in our 
pockets. To obviate this difficulty, I have prepared the annexed 
Abridged. Table of Natural Sines. Particular directions for its use 
are hereunto subjoined. A six inch ruler, 2 inches wide, will soon 
be made in Troy, N. Y.. containing the diagonal scale of equal 
parts, the chord line, and the abridged table of natural sines, carried 
out to sines of degrees and minutes. It will always be found at 
bookstores, where this Prodromus is sold, after the artist has com- 
pleted his instruments for making the ruler. 

ABRIDGED TABLE OF NATURAL SINES. 

At page 31, a table with this heading is printed. It was hurried 
into that form for cases where perfect exactness in the minutes 
should not be required. It was afterwards discovered, that by ex- 
tending the augments, degrees and minutes might always be calcu- 
lated and used, when books, containing full tables, were not at hand. 



DIRECTIONS 

For using the Abridged Table of Natural Sines, here inserted — also^ 
on the Pocket Ruler. 

1. If the sine of a whole degree, or of a half degree (30 minutes) 
is required, it is found against such degree or half degree (that is, 
such degree and 30 minutes.) 

2. If the sine of any number of minutes, less than 30 degrees, is 
required, proceed as follows : Find the number in the column of 
augments, against the last degree preceding the number of minutes, 
whose sine is required. Multiply this amount by the number of 
minutes, and add the product to the sine of the said preceding de- 
gree ; setting the right hand figure one place to the right of the sine. 
The sum will be the sine of the degrees and minutes required. 

3. If the sine of any number of minutes more than 30, is required, 
proceed thus : Find the number in the column of augments, against 
the last 30 minutes preceding the minutes whose sine is required. 
Multiply this augment by the number of minutes exceeding said 30 
minutes ; and add the product to the said sine of said preceding 30 
minutes — setting the right hand figure one place to the right of the 
sine. The sum is the sine of the degrees and minutes required. 

4. In an operation in trigonometry (where natural sines are used, 
see sec. 39) if the answer is in sines, find the degrees and minutes 
as follows : Find in the table the nearest degree or half degree, 
next less than is required. Subtract that sine from the said an- 
swer, and divide the remainder by the augment set against the de- 
gree or half degree ; which will give the additional minutes. These 
added to said nearest degrees and minutes, give the true degrees 
and minutes required. 

5. In most cases of ordinary practice, the subdivisions of a de- 
gree into six parts (10 minutes each) will be sufficiently accurate. 
Every sixth division (that is, every 10 minutes) requires a very 
simple application of the augments. Ten and 40 minutes require 
the augment once merely, without extending a figure to the right 
of the sine — 20 and 50 minutes require the augment doubled, and 
not extended to the right of the sine. 



ABRIDGED TABLE OF NATURAL SINES. 



3 = 


De- 


s. c-s. 


c-s. s. 


De- 


i'-3 


S - 


De- 


3. c-s. 


□-S. s. 


De- 1 


3 — 


< 1 


grees. 






grees. 


< 1 


< 1 


grees. 






grees.L- g 


291^ 


0.00 


0.00000 


1.00000 


90.00 




267 


23.001 .390721 


.92050 


67.00 


112 


291 


.30 


.00373 


.99996 


.30 


1 


266 


.30 


.39875 


.91706 


.30 


114 


291 


1.00 


.01745 


.99985 


89.00 


3 


265 


24.00 


.40674 


.91355 


66.00 


117 


291 


.30 


.02618 


.99966 


.30 


6 


264 


.30 


.41469 


.90996 


.30 


120 


291 2.001 


.03490 


.99939 


88.00 


8 


263 


25.00 


.42262 


.90631 


65.00 


121 


290 


.30 


.04362 


.99905 


.30 


11 


262 


.30 


.43051 


.90259 


.30 


123 


290 


3.00 


.05234 


.99863 


87.00 


13 


260 


26.00 


.43837 


.89879 


64.00 


126 


290 


.30 


.06105 


.99813 


.30 


16 


259 


.30 


.44620 


.89493 


.30 


128 


290 


4.00 


.06976 


.99756 


86.00 


18 


257 


27.00 


.45399 


.89101 


63.00 


130 


290 


.30 


.07346 


.99692 


.30 


21 


556 


.30 


.46175 


.88701 


.30 


133 


290 


5.00 


.03716 


.99619 


85.00 


24 


256 


23.00 


.46947 


.83297 


62.00 


135 


290 


.30 


.09585 


.99540 


.30 


26 


255 


.30 


.47716 


.87882 


.30 


137 


290 


6.00 


.10453 


.99452 


84.00 28 1 


253 


29.00 


.48481 


.87462 


61.00 


140 


289 


.30 


.11320 


.993.57 


.30 


31 


252 


.30 


.49242 


.87036 


.30 


141 


28S 


7.00 


.12187 


.99255 


33.00 


34 


251 


30.00 


.50000 


.86603 


60.00 


143 


288 


.30 


.13053 


.99144 


.30 


37 


250 


.30 


.50754 


.86163 


.30 


145 


287 


8.00 


.13917 


.99027 


82.00 


39 


248 


31.00 


.51504 


.85717 


59.00 


147 


287 


.30 .14781 


.93902 


.30 


41 


246 


.30 


.52250 


.85264 


.30 


151 


287 


9.00 


.15643 


.98769 


81.00 


43 


245 


32.00 


.52992 


.84895 


58.00 


153 


286 


.30 


.16505 


.98629 


.30 


46 


244 


.30 


.53730 


.84339 


.30 


155 


286 


10.00 


.17365 


.98481 


80.00 


49 


243 


33.00 


.54464 


.83367 


57.00 


157 


286 


.30 


.18224 


.98325 


.30 


52 


242 


.30 


.55194 .83389 


.30 


160 


235 


11.00 .19081 


.98163 79.00 


54 


240 


34.00 


.55919 


.32904 


.56.00 


162 


284 


.30 


.19937 


.97992 


.30 


57 


238 


.30 


.56641 


.32413 


.30 


163 


283 


12.00 


.20791 


.97815 


78.00 


59 


237 


35.00 


.57358 


.81915 


55.00 


165 


283 


.30 


.21644 


.97630 


.30 


62 


236 


.30 


.58070 


.81412 


.30 


167 


283 


13.00 


.22495 


.97437 


77.00 


64 


234 


36.00 


.58779 


.80902 


54.00 


170 


282 


.30 


.23345 


.97237 


.30 


66 


232 


.30 


.59482 


.80386 


.30 


172 


282 


14.00 


.24192 


.97030 


76.00 


69 


230 


37.00 


.60182 


.79364 


53.00 


174 


281 


.30 


.25033 


.96815 


.30 


72 


228 


.30 


.60876 


.79335 


.30 


176 


280 


1.5.00 


.25332 


.96593175.00 


74 


227 


38.00 


.61.566 


.78801 


52.00 


178 


280 


.30 


.26724 


.96363 


.30 


77 


226 


.30 


.62251 


.78261 


.30 


180 


279 


16.00 


.27564 


.96126 


74.00 


79 


225 


39.00 


.62932 


.77715 


51.00 


182 


276 


.30 


.28402 


.95832 


.30 


81 


223 


.30 


.63608 


.77162 


.30 


184 


277 


17.001 .29237 


.95630 


73.00 


83 


222 


40.00 


.64279 


.76604 


50.00il86 


277 


.30 


.30071 


.95372 


.30 


86 


220 


.30 


.64945 


.76041 


.30 


137 


276 


18.00 


.30902 


.95106 


72.00 


88 


218 


41.00 


.65606 


.75471 


49.00 


139 


275 


.30 


.31730 


.94832 


.30 


91 


216 


.30 


.66262 


.74896 


.30 


191 


274 


19.00 


.32557 


.94552 


71.00 


93 


215 


42.001 .66913 


.74314 


48.00 


193 


274 


.30 


.33381 


.94264 


.30: 96 


213 


.30 


.67559 


.73728 


.30 


195 


273 


20.00 


.34202 


.93969 


70.00 


93 


211 


43.00 


.68200 


.73135 


47.00 


197 


272 


.30 


.35021 


.9.3667 


.30 


101 


209 


.30 


.68835 


.72537 


.30 


199 


271 


21.00 


.3.5637 


.93358 


69.00 


103 


207 


44.00 


.69466 


.71934 


46.00 


201 


270| .30 


.36650 


.93042 


.30 


105 


206 


.30 


.70091 .71325 


.30 


203 


269 22.00 


.37461 


.92718 


68.00 107 


205 45.00 


.70711 .70711 


45.0C 


205 


2681 .30 


.38268 


.92388 


.30 110 


.30 


.71325 .70091 


.3C 





0^ Students are not to apply the above Table until they have 
studied sections 31 to 40, inclusive. 



CORRECTIONS TO BE MADE WITH THE PEN. 

Page 13, top line — "seventeen" change to " four." 

Page 31 — the augments are better on page vii. 

Page 80, sec. 141 — "foot" change to "inch." 

Page 91 — change places of the words "Top" and "bottom." 

Page 100, 3d line— after " Station" read "No. 10." 

Page 100, 3d line — instead of "moved up his instrument" read "went 

with the Targetman." 
Page 100, 19th line— for " Sec. 000" read " Sec. 227." 
Page 100, 23d line— for " 740" read "760." 
Page 103, 1st line — for " Station" read "two stations." 
Page 111, last of sec. 216 — between "the" and "ordinate" interline 

"square of." 
Page 114, sec. 224 — " chair" change to " bench." 
Page 122, sec. 253, middle line — strike out " the square of." 
Page 133, sec. 288, near the end — "to the root add" change to "from the 

root subtract." 



REFERENCES. 



The student is referred to last part of the book, where wood-cut figures are 
described, for continuations of several sections of importance. 

First is from section 198 to 200, extending and illustrating by wood-cut 
figures, the method of cnlcvbiting rail-road curves. 

Second is from sections 211 to 214, extending and illustrating by wood-cuts, 
the method of calculating ordinates, for offsets from secondary chord lines, 
one hundred feet each. 

Third is from sections 224 to 226, extending and illustrating by wood-cuts, 
the method of calculating excavations and embankments. 

This text-book is not limited in its object to students in surveying and engi- 
neering. Not more than 43 pages are, exclusively, devoted to them. In it 
will be found those rules and directions for calculations, which are essential 
to bverj correct student in Geography, Astronomy, and Natural Philosophy; 
also to all classes of readers, who wish to understand what they read. 

Teachers of female institutions, and of common academies, are requested 
to look over the contents, and consider the manner in which subjects are 
treated. 

Such institutions may omit — 

1. Practical Land Surveying, from page 47 to page 74. 

2. Running out Rail-Roads, from page 95 to page 111. 

All this treatise, excepting the above excepted 43 pages, should be studied 
and illustrated with practice by every student, who is presented to the public 
as tolerably educated. 

The common practice of introducing a system of elementary rules and de- 
monstrations, so to cumber a small practical treatise as almost to exclude the 
professed object, has always appeared to be absurd. Elementary treatises, 
executed in a style of excellence which cannot be surpassed, are to be found 
4 in almost every book-store. To them the learner is referred; and will not be 
taxed with the reprint, and expense of copy- right, for the sake of swelling a 
small work into a large one. Gibson's System of Surveying, for example, a 
work of almost 500 pages 8vo, contains less than 100 pages, which are devoted 
to the professed object of the work. It is hoped, that the learner will find 
nothing in this, which is useless in aid of his proposed object. 

With pleasure I acknowledge my obligation to State engineer Holmes 
Hutchinson, Esq., for his iiistructi/e explanations in answer to my numerous 

1 



inquiries for the last half dozen years. To engineer Wm. C. Young I am 
also indebted for much useful information on the construction of rail-road 
woiks; as exemplified and explained by him at the extensive works in Sche- 
nectady. For the latest and most approved method of rail-road surveying, in- 
cluding staking out, running curves, measuring for excavations, &c., students 
are referred to articles furnished by engineers Sargent and Evans. I vpill take 
this opportunity to say, that without exceptions, every practising engineer 
with whom I have had any intercourse during the existence of the Engineer 
department at this Institute (four years) has manifested a strong desire to aid 
its progress and extend its influence. 



CONTENTS. 



This table of contents is constructed for the convenience of examiners. As 
the by-laws of this Institute forbid all persons concerned in the instruction of 
students giving any opinion on the subject of their qualification, and as each 
board of examiners is made up of gentlemen who are wholly disconnected with 
the school; it was supposed, that few would be willing to devote sufficient 
time to each subject to extract the essential points in it. Therefore each item 
of the contents is made to contain what appears to be sufficient to give the 
student a fair clue to all that is expected from him. 

N. B. The word Explain, is supposed to be prefixed to every subject, in the 
imperative mode. In numbering sections, inclusive is understood. 



ARITHMETIC. 

No. of Secliona. 

1. Three elementary operations with number. Addition, Se- 

paration, . . . . . 1 to 5 

2. Notation, . , . . , 6 to 12 

3. Common characters, .... 13, 14 

4. Decimals, Addition and Subtraction, . . 16, 17 

5. Multiplication and Division, . . . 18, 19 

6. Bringing compound expressions to decimals, or decimals 

and integers, ..... 20 

7. Rule of three, ..... 21 

8. Roots and Powers, Square root, . . . 22 to 26 

9. Cube root, . . . . . 27 to 29 
10. Roots of higher powers than cube, ... 30 

TRIGONOMETRY. 

IJ. Angles and triangles, 

12. Trapezoid, and triangles between parallels, 

13. Sines, line of chords, degrees ia triangles, 

14. Geometrical trigonometry, 

15. Proportions of sides and angles of triangles, 
IG. Square root and rule of three applied to trigonometry, 

17. Natural Sines, .... 

18. Table of Natural Sines, 



31,32 

33(1,2) 

33 (3 to 6) 

34 

35 to 37 

38(1,2) 

39 

40 to 42 



No. of Sections. 

19. Operations in trigonometry when two sides of a rightrangled 

triangle are given, .... 43 

20. Also when one side and two angles are given, . . 44 

21. Also when two sides and an angle opposite to one of them 

are given, p . . . • 45 

22. Also when two sides and their contained angle are given, • 46 

23. Also if one leg of a right-angled triangle, and the sum of 

the other leg and hypothenuse, are given, . , 47 

24. Also when the three sides of a triangle are given. , 48 

MENSURATION. 

25. Parallelogram (or rectangle) triangle, polygon, and circle, 52 (1, 2, 3, 4) 

26. Periphery of a circle, diameter, and area, . . 52 (5, 6, 7) 

27. Length of an arc, sector, and segment, . . 52 (8, 9, 10) 

28. Area of an oval, .... 52(11) 

29. Superficies of a prism, cylinder, pyramid, and cone, . 52 (12, 13) 

30. Area of a parabola, and superficies of a globe, . 52 (14, 15) 

31. Solid contents of a cube and parallelepiped, . . 53 (I) 

32. Solid contents of a cylinder, prism, and wedge, . 53 (N. B.) 

33. Solid contents of a globe, pyramid, and cone, . . 53 (2, 3) 

34. Solid contents of the frustrum of a cone and pyramid by two 

methods, . . . . . 53 (4, 5, 6) 

35. Guaging by the double cone method, . . . 53 (7) 

36. Guaging by the common formula, . , . 53 (8) 

37. Tonnage of vessels, . . , , 53 (9) 

LAND SURVEYING. 

38. Four kinds of surveying, . . . 54 to 58 

39. Field surveying, . . . . . 59 to 62 

40. Preparations for farm surveying, putting needle, chain, tal- 

lies, &c., in order, . . . . 65 to 67 

41. Taking elevations and depressions by Kendall's tangent 

scale, ...... 71 

42. Fixing starting boundary and taking a course with the com- 

pass, . . . . , 74, 75 

43. Making offsets, ..... 78 

44. Taking distances and heights of objects not on the line, , 80 

45. Running a random line and making an extemporaneous cal- 

culation, . . . . . 81, 82 

46. The method of keeping the field book, . . . 89 

47. Plotting and triangular casting, . . , 91 to 95 

48. Reducing a field to a single triangle, , . , 96 

49. Trapezoidal method,' . . . , 100 to 104 

50. Road surveying, . . . . 110, 111 



No. of Sections. 



51. 



52. 

53. 
54. 
55. 

56. 
57. 

58. 

59, 

60. 
61. 

62. 
63. 

64. 
65. 

66. 



67. 



68. 
69. 

70. 

71. 

72. 
73. 
74. 



Harbor surveying by base lines on shore and intersections of 
lines of bearing, .... 

STATICS AND DYNAMICS. 
Distinctions between Statics and Dynamics, Hydrostatics 

and Hydrodynamics, 
Velocity of falling bodies, 
Weight of water compared with measure, 
Specific gravity of liquids as shewn by Baume's areometer, 

and his zero points, . . , . 

Taking specific gravity of solids for estimating their solidity, 
Method of demonstrating that water pressure depends on its 

height, ..... 

Velocity of spouting fluids increasing as the square roots of 

their heads above the point of effusion, 
Method of proving that atmospheric pressure holds water 

together in the liquid state, without which it would be- 
come vapor at 67 degrees of Fahrenheit, 
Method of determining how high to place the lower valve of 

a common pump, .... 

Gonatous forces applied to arches, bridges, &c., in general, 

MECHANICAL POWERS. 

Lever and its modifications, 
Inclined plane and its modifications, 

ARCHITECTURE. 

Pillars and parts of pillars, 

Orders of architecture, .... 

Miscellaneous structures, 



122 to 124 



127 to 130 

131, 132 

133 

134 
135 

136 

138 to 140 



141 

142, 143 
145 to 149 



151 
153 



154 to 156 

157 to 161 

164 



RAIL-ROADS, &c. 

Three kinds of survey — Extemporaneous, Preliminary, and Definite. 

Extemporaneous traverse across a mountainous district with 

barometer and compass, preparatory to a preliminary 

survey, .... 

Hutton's formula for calculating barometrical height, 
Gregory's formula. 
Latitude and longitude surveys of rail-roads and canals of 

great extent. 
Taking latitude, . - . . 

Longitude by Jupiter's moons, 
Longitude by lunar observations, 
Finding the breadth of a degree of longitude at any degree 

of latitude, ..... 





166, 167 




168,169 




170 


of 






171 to 173 


174, 


175 (note) 




176 


. 


177 



178 



6 

No. of Sections. 

75. Party for a preliminary survey of a rail-road, and the com- 

mencement of the survey with the compass, . 179 to 183 

76. Continuance of the survey with the level, and the method 

of keeping field notes, .... 184 

77. Definite rail-road survey, . . . 186 to 188 

78. Staking out, ..... 190,191 

79. Pencilling curves upon a plotted traverse, and finding the 

radius and central angle, . . . 198, 199 

60. Preparing for staking out the arc into 100 feet chords, . 200 to 202 

81. Finding the starting point and setting the compass upon 

the first point for deflexion from the tangent, . . 203, 204 

82. Preparing for, and running, on the general chord line and 

running ofisets (or ordinates) to the staking points on the 
arc, ..... 

83. Running and staking from one end of the arc, 

84. Making a hundred feet table of ordinates^ 

85. Changing the arc of a circle to the arc of an ellipse, 

86. Comparing rail-road curves, 

87. Convexity of the earth causing a falling below the level, 

EXCAVATIONS AND EMBANKMENTS. 

88. Finding the solid contents of earth to be excavated, &c., by 

six areas of a prismoid, .... 

89. Taking cross areas by applying the trapezoid, and triangles 

between parallels, .... 

CANALS. 

90. Displacement of water when boats are moved in narrow and 

wide canals, proving the advantage of wide canals, . 233, 234 

91. Boats constructed with a view to the principle of the in- 

clined plane and wedge, 

92. Technical terms applied to canals and locks, 

93. Calculating supply of water for a canal, 

94. Calculating the time of filling or emptying a lock, 

ROADS IN GENERAL. 

95. Shady trees, resting places, zigzag roads, dugways, 

96. Calculating the angle of friction of carriages on roads, 

97. Location of bridges, string pieces, and placing supports, 

WATERWORKS. 

98. Accelerating and retarding forces, 

99. Modifications of the two elementary forces, and their six- 

fold state, .... 



No. of Sections. 

100. Formula for pipes when velocity and quantity discharged 

are sought, ..... 286 

101. Formula for pipes when the diameter of the pipe is sought, 287 

102. Formula for open canals when velocity and quantity are 

required, ..... 288, 289 

103. Atmospheric pressure as influencing springs, &c., in pipes 

varying according to the height of mountains, &c., . 291 

104. Method of talung the height of the atmosphere at the point 

where it ceases to be dense enough to reflect light, . 292 

105. Calculating the time between sunset and setting of twi- 

light, as used in the above calculation, . . 293 

106. Aqueous vapor in fogs, clouds, and the five forms of regu- 

lar clouds, ..... 295, 296 

107. Three forms of disconnected clouds, . . . 297 

108. Taking the height of clouds, . . . . 298 

WATER-POWER, APPLIED TO MILLING, &c. 

109. Duty of an engineer, as to water power, confined to out of 

doors work, . . . . . 299, 300 

110. Calculating supply of water, . . . 301 to 306 

111. Calculating supply of water by a falling sheet of water, 

called the weir calculation, . . . 307 to 311 

112. Efficiency of water power at the bottom of flumes, . 312 to 314 

113. Water acting on undershot wheels, . . . 315 

114. Standard of efficiency of water-power taken from Troy 

mills, N. Y., . . . . . 317 to 319 

115. Efficiency of water-power, calculated from the square feet 

of the faces of mill-stones passing over each other, . 320,321 

116. Poestenkill mills as standards for calculating the general 

power of water, .... 322 

117. Calculating supply of water by the water-sheet pitch, or 

the weir method, ..... 323, 324 

118. Example of triturating surfaces of miU-stones, . 325, 326 

TOPOGRAPHY. 

119. Definition, and first steps in approximating surveys, . 327, 328 

120. A topographical survey in miniature, . . 329 to 331 

121. Extensive topographical surveys, . . . 332 to 334 

MATERIALS FOR CONSTRUCTION. 

122. Natural division of the subject, . . . 335,336 

123. General illustrations of inorganic and organic materials, 337, 338 

124. Geological alphabet, .... 339, 340 



No. of Sections. 

125. Arrangement of rocks geologically, or in regular series, . 341 

126. Exhibition and description of geological strata, 

127. Class of primitive rocks, .... 

128. Class of transition rocks, .... 

129. Class of loicer secondary rocks, 

130. Class of upper secondary rocks, 

131. Class of tertiarj/ earths, ...» 

132. Subordinate series, or the red sandstone group, 

133. Anomalous deposits, .... 

USEFUL ROCKS AND CEMENTS. 

134. Marble, freestone, flagging, &c., 

135. Millstone, grindstone, whetstone, 

136. Hornblende, granular quartz, argillite, . , 

137. Cements of lime, gypsum, «S:c., 

TIMBER MATERIALS. 

138. Oak timber, . . . . , 

139. Strength and durability of timber, 

IRON MATERIALS. 

140. Three kinds of iron, . . . , 

141. Application of kinds of iron, j . . . 



MATHEMATICAL ILLUSTRATIONS. 

Before studying this treatise, students must have been sufficient- 
ly exercised under teachers' dictum, with whole numbers and decii 
mals, in Addition, Subtraction, Multiplication and Division, and the 
Rule of Three. Nothing more is required ; neither is it profitable 
to detain students with Compound Arithmetic, Vulgar Fractions, or 
Algebra, until they are made acquainted with the most useful parts 
of the Mathematical Arts — particularly the general applications of 
superfices, solids, and trigonometry, to common business concei'ns. 
This I assert ; and my assertion is founded on forty years' experi- 
ence. And this rule applies to merchants' clerks and others, whose 
operations are merely arithmetical. The power of numbers, once 
understood, applies to all cases alike. Therefore the mere land 
surveyor makes a better book-keeper by one month's practice, 
than the student in mere book-keeping does in a year. The reason 
is manifest. The surveyor takes, necessarily, a scientific view of 
the power of number; while the student in book-keeping takes a 
parrot-Uke rotine of artificial forms. The former is governed by 
sound reason--the latter is led blind-fold by authority. He is a 
mere machine — but the mathematician is disciplined as an intellect- 
ual being. 

Alcohol and Algebea are Arabic names. Alcohol is a power- 
ful agent, of vast importance. But its abuses render it a curse. 
Algebra is a powerful entering wedge in Mechanics, and of great 
importance in the concise expressions of valuable formulae. But a 
kind of affectation of technical learning has so far obscured the 
mathematical arts with algebra, as to render it an absolute nuisance. 
In this little treatise, algebraic formulce are translated into fair 
English. Biot, (a most distinguished French Philosopher,) after 
thirty years' experience, as teacher in the algebraic mode of ex- 
pression, prepared a System of Natural Philosophy, totally divested 
of all such technical obscurities. His authority, supported by sue 
cess, has, in a great measure, revolutionized the course of mathe- 
matical learning in France. 

2 



10 



ELEMENTARY OPERATIONS WITH NUMBERS. 

Sec. 1. The science of numbers, called Arithmetic or Mathema- 
tics, may be resolved into three elementary operations : Addition, 
Separation and Notation. 

Addition. 

Sec. 2. This operation consists in uniting individuals into groups, 
masses, or sums; as the arranging of 100 soldiers into a group, 
called a company — uniting 196 pounds of flour in a mass, called a 
barrel — uniting the value of 100 cents into a silver coin, called a 
dollai' — or uniting in one sum a sufficient number of feet of plank 
for laying a floor in a room of given length and breadth. 

Sec. 3. When several additions are performed by one operation, 
w^e distinguish this modification of addition by the descriptive name, 
Multiplication: as $12 paid to each of 7 laborers must be added 
seven times to find out the whole sum to be paid, thus : 12+12+12 
+12+12+12+12=84. But we may learn by rote to add thus : 
seven times twelve equals 84. This rotine metjiod of adding is 
called Multiplying. 

Separation. 
Sec. 4. This operation consists in taking part from the whole ; 
or separating smaller portions of masses, or numbers, from the 
larger. Individuals of groups are separated from each othei*, or 
distributed into smaller groups. The operation is called Subtrac- 
tion, or Division, according to its peculiar application. All results 
may be produced by Subtraction; but Division is more expeditious 
when a separation into numerous parcels or parts is required, and 
particularly when the proportional parts are to be ascertained, 
from given data, for proportional separation. 

If 1281 dollars are to be equally divided among 61 laborers, we 
divide the dollars by 61, by a tabular rotine, called Division, thus: 
61)1281(21— giving $21 to each. 
122 

61 
61 

00 



11 

Sec. 5. The same result will be produced by perpetually sub- 
tracting 61 from 1281, and counting up the number of subtractions, 

thus : 



§1281 


854 




427 


61 1st. 


61 
793 


8th. 


61 15th. 


1220 


366 


61 2d. 


61 


9th. 


61 16th. 


1159 


732 


305 


61 3d. 


61 


10th. 


61 17th. 


1098 


671 


244 


61 4th. 


61 


11th. 


61 18th. 


1037 


610 


183 


61 5th. 


61 


12th. 


61 19th. 


976 


549 


122 


61 6th. 


61 

488 


13th. 


61 20th 


915 


61 


61 7th. 


61 


14th. 


61 21st. 



854 carried up. 427 carried up. 00 

As 61 can be subtracted 21 times from $1281, each of the 61 
laborers will have $21, as given by dividing by 61. 

Notation. 

Sec. 6. The operation of setting down or recording numbers. 
Numbers are expressed by Roman letters, or by Arabic figures. 
But Roman letters are never used in the process of calculations. 
They are very convenient for expressing the numbers of large 
divisions which are to be subdivided : such as Classes of Plants, 
expressed in Roman letters, which are subdivided into Orders, and 
expressed in figures. 

Sec. 7. The Roman letters used for expressing numbers are, I 
for one, V for five, X for ten, L for fifty, C for a hundred, D for 
five hundred, M for a thousand. When these letters are joined in 
a horizontal row from left to right, with the smallest valued letter at 



12 

the right, that is added to the larger. But if the smaller valued let- 
ter is set on the left of the larger, it is subtracted. Thus, VI stands 
for six, and IV for four; XI for eleven, IX for nine; LX for sixty, 
XL for forty; CX for one hundred and ten, XC for ninety. No 
letter, however, but I, X and C, is used by us in this manner as a 
subtrahend. 

Sec. 8. The figures are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. When join, 
ed in a horizontal row, they begin their value with the right hand 
figure ; and the figure, added on its left, stands for ten times its 
value when alone : the next for one hundred times as much : the 
next for one thousand times, and so on. In the notation of figures 
they are pointed off in threes, from right to left. 

Sec. 9. The element of the first three numbers is unit — of the 
second three, is thousand — of the tliird three, is million — of the 
fourth three, is billion — of the fifth three, is trillion — of the sixth 
three, is quadrillion — of the seventh three, is quintillion — of the 
eighth three, is sextillion — of the ninth three, is sepiillion — of the 
tenth three, is octillion — of the eleventh three, is nonillion — of the 
tivelfth three, is decillion — of the thirteenth three, is undecillion; 
and so on, indefinitely, following the Latin numerals, with the same 
Anglicising termination. 



01 


CO 

1 § 


1 


1 rA 




1 


1 


1 


1 


n 


•■^ 


00 


2 


. 










o 


a, ^ 


^ CI 


^ s 


Cnii W 


d, 


^ 


v< 


tJL, 




o -S 


O O 


o .2 


° o 


eds 

f— 

ions. 


° ^ 


o 


° =^ 


.^ 


m 1 <u 


EC 1 r5 


w 1 ;=! 


m 1 -^ 


CO 1 H 


CO 1 52 


CO I a 




Tj ■^ 


^ ^ 


id red 
s of— 
odeci 


-o ^ 


^ O 


'^ s 


t: o 


UJ 


<D ^ ■•:; 


<u 'vL " 




CO iJ-,--:! 


o OL o 


QJ ti -^ 


'TJ 


§ § § 

^ Ba 


!-^ o '^ 


•- O =2 




f- O r=3 


s- O -^ 


C 


o CO o 


hund 
tens 
Deci 


'^ »'s 


hund 
tens 
Octil 


hunc 
tens 
Septi 


a 


s n '-H 
-a B^ 


S C D 

M BQ 


Sen 
i B\^ 


3 = 1^ 
M B'^ 



17 



16 



15 



14 



13 



12 



11 



10 



4, 8 6 7, 4 3 9, 2 4 5, 6 1 7, 6 3 5, 9 2 6, 7 3 5, 7 2 4, 



!^ o 

^1 



..I. . 


tji CO 


a 


a 




i 


1 


a 




o » 

CO I O 


° § 
^ 1 'J=i 


o 

^ 1 i 


o 

00 1 




o 
<» 1 


hundreds o 
tens of — 
Thousands 


o 

rn 


1 


^ -^ 


^ ^ 


'^ J 


• 


'TS w 


'XS 




q; V- rs 


Q c+i, -r; 


2%M 


q; eJ, 


r^ 


o tJL< c 


<v 


uL 


•-H o -Ti 


^^^ 


!h O 


.o 


!- O O 






^ m.S 


^ -"S 


■M r/, 


hunc 
tens 
Milli 


T3 
G 


§ G 3 


Sea 


^Bh^ 




cS 


G G 

B t5 



6 



78 9, 63 5, 24 6, 85 7, 84 6, 45 6, 78 9, 567 



13 

Sec. 10. This example is to be read as follows: seventeen quin- 
decillions, eight hundred and sixty-seven quadridecillions, four hun- 
dred and thirty-nine tredecillions, two hundred forty-five duodecil- 
lions, six hundred seventeen undecillions, six hundred thirty-five 
decillions, nine hundred twenty-six nonillions, seven hundred thirty- 
five octillions, seven hundred twenty-four septillions, seven hundred 
eighty-nine sextillions, six hundred thirty-five quintillions, two hun- 
dred forty-six quadrillions, eight hundred fifty-seven trillions, eight 
hundred forty-six billions, four hundred fifty-six millions, seven hun- 
dred eighty-nine thousands, five hundred fifty-seven. 

Sec. 11. Gregory says, this is the method of notation adopted 
in France, Germany, &c., which he prefers. But the English often 
proceed by repeating millions ; as millions of millions of millions of 
millions, &c. Take the following example of using million as a 
substitute for billion, trillion, &c. : 

^ 5 4 3 2 1 

8,054,327,923,578,263,523,127,354,687,128,354,627. 

8 millions of millions of millions of millions of millions of millions, 
654,327 millions of millions of millions of millions of millions, 923,. 
578 millions of millions of millions of millions, 263,523 millions of 
millions of miUions, 127,354 miUions of millions, 687,128 millions, 
354,627. 

Sec. 12. The language of number is divided into Cardinal and 
Ordinal adjectives. As one, two, three, four, five, six, &c., are 
cardinal names. First, second, third, fourth, fifth, sixth, &c., are 
ordinal names. Ordinal names, or adjectives, are changed to ad- 
verbs by adding ly — as fifthly, tenthly, twenty-eightly, &c. 

Common Characters. 

Sec. 13. A point, or period, as a character, is important in no- 
tation. It should always stand at the right of a horizontal series of 
integers, if decimals are appended. Thus : 62754 inches, and 726 
thousandth of an inch, is thus expressed, 62754.726. In enumera- 
ting we begin at point, and say, units, tens, hundreds, thousands, 
tens of thousands, on the left — and tens, hundreds, thousands, on the 
right. We therefore read this example thus : sixty-two thousand, 
seven hundred and fifty-four inches ; point, seven, two, six. 



14 

Sec. 14. Double points, or colons, are used in the Rule of Three, 
thus : if $70 buy 9 acres, what will $926 buy? Is set down, $70 : 
9a.:: $926. The first single colon is to be read " is to ;" the last 
is read "/o;" and the double colon is to be read "so is." There- 
fore the true reading of the above example is, as $70 is to 9 acres, 
so is $926 to 119/^ acres— same as 70 : 9 : : 926 : 119 ^V 

Right Cross +, [called plus,) signifies that the number following 
it is to be added to something preceding, as 267+31, shews that 
31 is to be added to 267, making 298. 

Oblique Cross X> signifies that the number following it, is to be 
multipUed into something preceding, as 124x12, shews that 12 is 
to be multiplied into 124, making 1488. 

Horizontal line — , (called minus,) signifies that the number fol- 
lowing it is to be subtracted from something preceding, as 729 — 36 
shows that 36 is to be subtracted, leaving 693. 

Horizontal dotted line ~^, signifies that the number following it, 
is to be used as a divisor for dividing some dividend that precedes 
it, as 806-4-26, shews that 26 is to be used as a divisor for the divi- 
dend 806, making 31. This dotted line is often omitted, and the 
figures set down like an impi'oper vulgar fraction, thus, y^. 

Parallel lines =, signify that the number or numbers following 
them, equal something preceding, as 17x5x4=:340. 

Inverted figure seven \/, signifies that what follows it, requires 

the square root to be extracted. Figure 3 over it imphes that the 

a 
cube root should be extracted, &c., as \/27 requires the cube root 

to be extracted, making 3. 

N. B. Students must be exercised in notation, until they are 
familiar with the right application of the characters, and arrange- 
ment of figures. 

Decimals. 

Sec. 15. Decimals are parts of mtegers (whole numbers,) point- 
ed off" to the right by a period. The first figure expresses tenths of 
an integer, the second hundredths, and so on, diminishing ten fold 
at every figure. Several figures of decimals express a general 



15 

proportion of an integer, as 9.6345 inches expresses nine inches and 
6345 ten thousanths of an inch. But the most common as well as 
most convenient mode of expression is, to read off the figures 
separately after the point — thus : nine, point, six, three, four, five. 
In reading off Natural Sines, or Logarithms, this method is always 
to be adopted. 

Addition of Decimals. 

Sec. 16. Decimals, or integers and decimals together, are al- 
ways added like whole numbers. But in setting them down for 
adding, points must be set under each other in a column ; and let 
the figures on both sides of each point stand at uniform distances. 
This will cause some lines to project further to the right and left 
than others. Of course care is required in footing up the figures in 
their proper columns. 

Examples in Addition of Decimals. 

76543.6201 63421.6203 
360.2 9. 

2.76423 243.67861 

4239.621 27. 

7.3 3.00001 

36012.00034 789.9 



117165.50567 64493.19892 

Subtraction of Decimals. 

Sec. 17. Decimals, or integers and decimals together, are al- 
ways subtracted like whole numbers. But in setting them down 
for adding, the point in the subtrahend must be set directly under 
the point in the minuend ; and the figures each side of the point 
stand as directed in addition. 

Examples in Subtraction of Decimals. 

63967.9342 80001. 

201.7 2.0096 



63766.2342 79998.9904 



16 



Multiplication of Decimals. 

Sec. 18. Decimals, or decimals and integers together, are always 
multiplied like whole numbers. But after multiplying, care is re- 
quired in placing the point between the integers and decimals. 
The rule is, to point off for decimals just as many places of figures, 
as are pointed off in both the multiplier and multiplicand ; leaving 
for integers all the rest, if any. 

Examples of Multiplication of Decimals. 

Diameter of a circle, 643.231 inches. Formulae for circumfer- 
ence, 3.1416 inches. 

643.231 (operation 
3.1416 omitted.) 



2020.7745096 



Four decimal places in the multiplier, and three in the multipli- 
cand, require seven to be pointed off. Therefore the circumference 
is 2020 inches and 77 hundredths, if hundredths come near enough. 
But if exactness is required, say 2020 inches, and point .7745096 
decimals of an inch. 

Area of a square embracing a circle is 5402064.201 

Formula for reducing a square to a circle, .7854 



4242781.2234654 



Four decimal places in the multiplier, and three in the multipli- 
cand, require seven to be pointed off. Therefore the area of the 
circle is as above — that is, 4242781 square inches, and 22 hun- 
dredths, if hundredths come near enough. Or, point .2234654 deci- 
mals of an inch. 

Division of Decimals. 

Sec 19. Decimals, or decimals and integers together, are al- 
ways divided hke whole numbers. But, after dividing, care is re- 
quired in placing the point between the integers and decimals. 
The rule is, to point off for decimals in the quotient so many places 
of figures, as to make the number pointed off in the divisor and quo. 



17 

tient, just equal tjie number pointed off in the dividend alone. But 
if there are not as many pointed off in the dividend as in the divi- 
sor, cyphers must be added to the dividend (beyond the point) to 
make the number equal. And if an equal number does not carry 
on the decimals of the answer far enough for the required exact- 
ness, more may be annexed — always applying the rule for pointing 
off, as before stated. 

N. B. We often proceed by adding decimals, in carrying on 
the decimals in the quotient to greater extent than those of the divi- 
dend will admit. In such cases, all the added decimals must be 
counted as if they had previously been added to the dividend. 

Examples in Division of Decimals. 

The area of a circle is 4242781.2234654 inches; divide by the 
formula .7854, which will give the area of a circumscribing square : 

.7854)4242781. 2234654( (operation 

5402064.201 omitted.) 

•« ' 

Here are four places pointed off in the divisor and seven in the 
dividend. Now there must be pointed off in the quotient, which 
added to those of the divisor (being all the divisor) equal those of 
the dividend. The answer then is, that the circumscribing square 
made about the given circle, contains 5402064 square inches, and 
point .201 decimals of a square inch. 

Sec. 20. Bring compound expressions to Decimals. In all cases 
where measures, weights, or values of any kind, are required to be 
brought into decimal expressions, make a vulgar fraction express- 
ing the proportions. Then divide the numerator by the denomina- 
tor ; adding cyphers to the numerator as far as may be required. 

Bring £16 7s. and 4d. to the decimal of a pound: 7s. and 4d.= 
88d. A penny is the 240th of a pound. Then seven and four- 
pence is ^Vo of a pound— 240)88.0000(.3666. Answer is £16. 
3666. Note: this results in a circulating decimal ; as it will for- 
ever give 6, and is ever approximating the truth. 

Rule of Three. 

Sec. 21. When a value is set upon one article, it is self-evident 
that the value of any number of articles may be found, by multiply. 



18 

ing the value of one by the number whose whole value is required. 
As, if one square foot of ground costs 60 cents, and a house lot may 
be had at the same price per foot, which contains 1800 feet, it is 
manifest that the whole lot will cost 1800 times 60 cents, or 1080 
dollars. 

This stated, according to the form of the Rule of Three, will stand 
thus : 

1 : $0.60 :: 1800 : $1080.00. 

Here the second and third numbers are to be multiplied together 
to produce the fourth for the answer. But if the first number is 
more than one, the fourth number must be divided by it ; or it will 
be too great. 

Therefore, if four feet should cost but 60 cents, then the answer 
would be four times too much, and must be divided by 4 — ^thus : 

4 : $0.60 : : 1800 : $270.00 
.60 



4)1080.00 
270.00 

Therefore, these arrangements and operations meet all possible 
cases, where we can say — if the first number gives, produces, pur- 
chases, &c., the second, what will the third give, produce, purchase, 
&c. The second and third being inter-multiplied, always give the 
answer, on the supposition that the first number contains but one 
of the articles under calculation. If it contains more than one, the 
answer must be reduced to the truth, by dividing by it. 

Example. If the excavation of 27 yards of earth cost $17.25, 
what will the excavation of 77 yards cost ? 

27 : 17.25 : : 77 : 49.19 

77 



)1328.25(49.19 

If-one yard of excavation should cost the $17.25, the 1328.25 
would be the answer without being divided by 27. But as the 
whole 27 yards cost but $17.25, that answer would be 27 times too 
much, if not reduced by this divisor. 



19 



ROOTS AND POWERS. 

Any number is called a root, when considered in relation to its 
powers ; and its powers are estimated by the times it is multiplied 
by itself. As 2x2=4 is 2d power — 2 x 2x2=8 is 3d powdr — 2x 
2x2x2=16 is 4th power— 2x2x2x2x2=32, 5th power, &c. 

There cannot be a real exhibition of any power above, the 3d, or 
cube. But it is often necessary in a calculation to advance, ideally, 
into the higher powers. 

No student is qualified for entering upon the study of the Mathe- 
matical Arts, without a good knowledge of the square root and of 
the cube root. These are made as familiar, in this treatise, as any 
diligent student can desire. 

Hutton's concise method for approximation in all the higher pow- 
ers, (sufficient for ordinary practice,) is given at the end of the 
roots. 

SQUARE ROOT. 

Sec. 22. A distinction between the Squdre and the Root must be 
understood by students. A root multiplied by itself produces the 
square : as the root 4 multiplied by itself, produces the square 16. 
When we use the word root, with or without the word square pre- 
fixed, we mean the root only. When we intend to express what is 
produced by multiplying the root by itself, we use the word square 
only. 

Operations under the head of square root are divided into Invo- 
lution and Evolution. It is an operation in Involution to multiply 
10 by 10 and produce 100. It is an operation in Evolution, hav- 
ing 100 given, to find the number which multiplied by itself will 
produce the given hundred. The operation is called extracting 
the root. 

iNvoLirnoN OF THE Sqtjare Root. ' 

Sec. 23. This process requires no instruction. It is simple mul- 
tiplication, limited to multiplying the same number by itself; as 
twelve times twelve give one hundred forty.four. 



20 



Evolution of the Square Root. 

Sec. 24. This operation, when applied to squares, whose roots 
are found in whole numbers, within the limits of common multipli- 
cation tables, is a mere mental operation requiring no figures. As 
the root of 144 is 12— of 121 is 11— of 100 is 10— of 81 is 9— of 
64 is 8— of 49 is 7— of 36 is 6— of 25 is 5— of 16 is 4— of 9 is 3— 
of 4 is 2 — of 1 is 1 . No intermediate case is found among these 
examples, which can be thus easily and mentally solved. But all 
the intermediate squares will give fractions of numbers in their roots. 
And all higher numbers present analagous difficulties, and still 
more complicated. Hence Evolution, or the Extraction of the 
Square Root, requires study and method. 

Sec. 25. Directions for extracting the Square Root. As the 
number of figures in the root will always be equal to the number 
of pairs in the square of it, the pairs are pointed off from right to 
left. If an odd figure remains at the left end of the line, this alone 
will give one point, and of course stands in the place of a pair. 
Therefore we can foresee the number of figures which the root will 
contain, as soon as the square is pointed off. The last pair, or 
point, on the left, gives a number, which always contains more than 
half of the whole root ; consequently that point in the square ex- 
ceeds, in breadth, all the rest of the area of the square. 

Sec. 26. Take for example 45623.0000 square chains. These 
chains we wish to lay out in a square piece of ground. One side 
of this square lot will be the root of these square chains. There- 



21 

fore it is a side of the lot we are in search of. 

the first point at the left 

will be 2, and this must 

have two figures at the 

right of it, besides the 

decimals, the nucleus 

square A must be 200 

chains on each side. 

The slip B, 10— slip c, 3 

— slip d, .5 — slip e, .09. 



As the square of 

e .09 

d .5 



A 



Operation. 

45623.0000(213.59 
4 



41)56 
41 



w 423)1523 
1269 



T 4265)25400 
21325 



s 42709)407500 
384381 



-10. 



■200. 



213.59 

is the length of 

one side of the 
lot, or the root 
of the given 
square. 



23119+ 

N. B. The teacher must shew the student to perform similar 
operations frequently ; then he will be prepared to understand this 
illustration. 



CUBE ROOT. 

Sec. 27. The same distinction between Involution and Evolution 
must be understood by students under the cube root as that de- 
scribed under the square root. 



22 



Evolution of the Cube Root. 

Sec. 28. As the number of figures in the cube root will always 
be equal to the number of tripple figures in the sum given for ex- 
traction, the given figures are pointed off" in threes from right to 
left. If two or one remain, a separate point is also made of such 
remainder. Therefore we can foresee the nuinber of figures, 
which the root will contain, as soon as the given figures are point- 
ed off", as mentioned under the square root. The cube root of the 
last point on the left will exceed that of all the remaining points; 
as all the rest must stand on its right, and, of course, diminish ten- 
fold in value. (Cyphers, equalling the number of points, may be 
annexed to it.) This cube root will be the length (or linear extent) 
of the nucleus cube; upon three sides of which all that remains to 
be extracted is placed in layers. Therefore the operation is to be 
so performed, as to give the thicknesses and superfices of all outer 
layers, and of the fiUing-in corners. 

Directions for Extracting the Cube Root. 

Operation. 

Cube 75686967(423 Root. 
64 



5044)11686 
10088 



532989)1598967 ^ 
1598967 

Auxiliary operation carried on with the above. 

1st. 4x4x4=64 square of the nucleus cube. 

2d. 4X4=16 area of each face of the nucleus cube. 16x3= 
48 area of the three faces, being the trial divisor. 

48)116(2 thickness [last figures left ofF, 86.] 
96 

20 



23 

4-{-4-|-4=12 length of the three parallelepipeds. 
Length of the cube — 2 set to the right, as of less value. 



122 
Width same 2 



244 area of all the cube parallelepipeds. 
Area of the faces 48 



5044 true divisor for second figure. 
3d. 42x42=1764x3=5292 area of the three faces. 

5292)15989(3 [off 57] 42+42+42=126 length of parallelo- 
thickness 3 [pipeds. 

1263 
3 

So on like the rest above, &c. 

Sec 29. Explanation. Students must inspect a model, while 
examining this calculation. If no wooden model is prepared, pro- 
ceed thus : Cut a potatoe cube, about one inch. Cut three potatoe 
slabs one fourth of an inch thick, whose area (each) will precisely 
cover three of the sides of the cube — pin them on their respective 
sides, so that when on, the whole shall be an enlarged cube ; but 
with unfilled corners. Fill the three long corners with potatoe 
parallelopipeds ; and fill the place where all the pieces leave an 
open corner, with a cube of the same thickness of the parallelo- 
pipeds. 

By the operation annexed the nucleus cube is obtained by trial ; 
that is, by multiplying the nearest figures, until the nearest is found 
within the point (75 in this case,) as 5x5x5=125 — this being too 
high, say 4x4x4=64 — this being the nearest below 75, consider 
4 the meeisure of the nucleus cube. 

After bringing down the next point, divide by a trial divisor, 
made up of the three areas to be covered with slabs ; the quotient 
will be their thickness. The same thickness will equal the breadths 
of the parallelopipeds and of the corner cube. Their united length 
multiplied by their breadths, and added to the aforesaid areas, give 
the whole area. 



24 

Explanations. This area is the true divisor of the 11686 in this 
example. In adding the area of the three faces, of the parallelopi- 
peds, and of the corner cube, their respective areas must be arrang- 
ed from left to right according to their values. In this case, 48 is, 
truly, 4800—12 is 120—2 is unity. Therefore they add thus: 

2 
120 

4800 



4922 



But the areas of the parallelepipeds and corner cubes being ob- 
tained by a joint multiplication, they form 244 before 48 is added. 
This explains that position of them. 

Students will perceive that the area of but one face of the paral- 
lelepipeds and corner cube are calculated. This is on account of 
their mitred character ; as the middle of each side, only, is reckon, 
ed in their areas. (See figure.) The lengths of the two dotted 
lines give the area, which is equal to the whole of one side ; there- 
fore one side, only, is taken. 

Sec. 30. Hutton'^s approximating method for the higher poioers. 
Make several trials, until the number for the root, above and be- 
low the true root, is found. Involve the number below as the root 
to find its cube, &c., according to the power required. Call this 
root an assumed root — this power an assumed power. 

Then say by the rule of three, as the sum of the given, and 
double the assumed cubes is to the sum of the assumed, and double 
the given cubes, so is the assumed root to the root required. 

The fourth power being 67543, to find the Root. 

17x17 three times gives 83521. This is above. 16x16 three 
times gives 65536. This is then the first below. 



25 

First Operation. 

The assumed power, or the power of the assumed root 16, is 65536. 

65536 assumed. 67543 given. 
2 2 



131072 
67543 given. 


135086 
65536 assumed power of root 16. 


198615 : 


200622 :: 16 : 16.1616 




Second Operation. 


67038.79 
2 


67543 
2 


134077.58 
67543. 


135086 
67038.79 assumed power of 16.164- 


201620.50 : 


202124.79 :: 16.16 : 16.2 




TMrd Operation. 


68874.75 
2 


67543 
2 


137749.5 
67543. 


135086 
68874 assumed power of 16.2 


205292.5 : 


203960.75 :: 16.2 : 16.09 



As far as these operations are carried on, the nearer they ap- 
proach the true root. But these alternate above and below the 
truth ; and the medium is not to be considered as the truth. When 
the root is supposed to be nearly found, the proof is shewn by invo- 
lution. 

But all evolutions of the powers above the cube are long and 
tedious. 



26 



TRIGONOMETRY. 

Sec. 31. Angles, or corners, are: 

1. Right angle, square corner. 

2. Obtuse angle, larger than a square corner. 

3. Acute angle, smaller than a square corner. 

Sec. 32. Triangles, or three-sided figures : 

1. Right-angled triangle, having one right angle. Its perpendi- 
cular line may be called the vertical leg — its bottom line may be 
called the horizontal leg. 

2. Obtuse-angled triangle, having one obtuse angle. 

3. Acute-angled triangle, having all the angles acute. 

4. Isosceles triangle, having two of the sides of equal length ; 
consequently, having two equal angles. 

5. Equilateral triangle, having all the sides of equal length ; 
consequently all the angles equal (just 60 degrees each.) This 
triangle necessarily includes the isosceles triangle. 

6. Scalene triangle, having no two sides of equal length. The 
three first named kinds of triangle may be scalene. 

Sec. 33. Five miscellaneous items, which should here be noticed 
by the student. They will be farther illustrated. 

1. A trapezoid is any four-sided figure with only two parallel 
sides. Any figure bounded by right lines may be cut into trape- 
zoids by latitude and departure lines [to be explained further on.] 
And the superficies of each trapezoid may be found by adding the 
parallel sides, multiplying their sum by their distance from each 
other, and halving the product. 

2. Equal triangles. Two lines are parallel, when equi-distant 
from end to end ; and all possible triangles, made between them on 
the same base, contain equal superficies, or areas. Any figure 
bounded by right lines may be reduced to a single triangle by the 
application of this principle. [To be explained further on.] 

3. Degrees of a circle, are 360 ; each quarter or quadrant con- 
taining 90 degrees. The sine of an angle is a line let fall from one 
end of the arc of the angle, perpendicularly upon the opposite side. 
As the line a c is the sine of the angle e. 



27 

4. Line of cJiords, is a graduated line connecting the two ends of 
a graduated quadrant by a chord Hne. It is made by setting one 
foot of the dividers at a, and extending the other foot to each degree 
on the arc, and turning it down to the chord line. The chord hne 
is used in plotting or projecting, when geometrical calculations are 
to be made. 

5. A triangle contains 180 degrees; for ahalf circle contains 180 
degrees, which is represented by the serai-circle g h i. By inspec- 
tion, without taking the steps of a demonstration, the reader will per- 
ceive that the angles a and a are equal, also c and c, also e and e. 
Now as the angles ace below the line g i, are measured by the 
semi-circle g h i, it follows that the angles a c e in the triangle above 
the line g i, may be measured by the same ; consequently, contains 
180 degrees. The same elucidation may be given of all forms of 
the triangle. 

6. In every right-angled triangle, the long oblique side is called 
the hypothenuse. The other two sides are called the legs. And it 
may here be shewn the student, that the square of the two legs is 
equal to the square of the hypothenuse. This is a very important 
principle in practical trigonometry. 

Sec. 34. Geometrical trigonometry gives all the sides and all the 
angles of a triangle, if two angles and one side, or two sides and one 
angle, are previously taken by the proper measures and observations. 

From the following exemplification of this proposition, a reader of 
ordinary ingenuity, with no previous knowledge of trigonometry, 
may make all the applications which the following treatise requires. 
Draw the lines and angles given, then finish out the triangle in the 
only way in which it can be completed, without any random opera- 
tion. The sides 70, 80, in the figure, and the angles at them are 
given. Draw the given line fifty feet, calling any division of a scale 
a foot. Strike the arc 80, d, with a radius of 60 degrees, taken 
with the dividers from the line of chords on Gunter's scale. Take 
70 degrees, being the angle at 70, from the same line of chords, and 
set it off, on the arc, which will extend to d. Draw the line from 70 
through d indefinitely. Then with the sweep of 60, again strike the 
arc 70, /, and set off 80 degrees upon it, being the angle at 80, 
which will extend to ef. Draw the line from 80 through / indefi- 
nitely, and it will cross the line which was drawn through 70, at g. 



28 

Where these lines intersect each other, is the true place for the other 
angle. Measure the two new sides by the same scale by which the 
given line was laid down, and you have all the sides. Add the two 
given angles together, which will make 150 degrees. Subtract this 
sum from 180, the degrees always contained in a whole triangle, 
and the remainder will be 30, the degrees of the new angle at g. 

Sec. 35. Proportions of sides and angles. The angles of all tri- 
angles are proportioned to their opposite sides ; therefore when two 
angles and a side, or two sides and one angle, are given, the other 
angles and sides may be found by the common rule of proportion, 
(rule of three.) But sines (as represented in number 3 of Sec. 33, 
a c) are used instead of degrees. These sines stand in tables, ac- 
cording to their actual lengths — calling the sine of 90, one. 

Sec. 36. Trigonometry is a most sublime application of Mathe- 
matics. By it we learn the distances and movements of celestial 
bodies, which are millions of miles distant from us. And by it, 
also, we ascertain powers and movements, essential to our daily 
duties and comforts. Trigonometry, after all, owes its mighty pow- 
ers to those laws, by which we ascertain two unknown sides of a 
three sided figure, by having but one side measured. The adept 
in mathematics knows, that the science requires that its applicants 
should be able to find one side from two, when the triangle has a 
square corner in it — ^or so that the sharpness and bluntness of a 
corner, made by the meeting of two sides, is necessary to most cal- 
culations. But these may be ranked with other principles and ope- 
rations ; the peculiar essentials being those just stated. 

Sec. 37. A corner (always called an angle in treatises,) is best 
measured by a circle, struck around it with compasses, (usually 
called dividers,) with one foot standing exactly in the angle. The 
piece of the circle between the lines making the corner, is the 
measure of the angle. Every circle has always been divided into 
360 degrees. Therefore, if one quarter of the circle, or 90 degrees 
of it, is included between the lines, the angle is a square corner, or 
90 degrees — so of any other-sized angle. The circle is preferred 
as a measure ; because a circular instrument for measuring has the 
most universal application. Students must learn the use of the 
compass, quadrant, sextant, &c., by actual shewing only. 



29 

Sec. 38. The two great principles applied to trigonometry are : 

1. Application of the square root to the theorem, that in every 
triangle which has one square corner in it, (called a right-angled 
triangle,) the square of the slant-side (called hypothenuse) is equal 
to the square of both of the other sides — they being first squared 
separately and then added together. 

2. Application of the rule of three to the theorem, that sides of 
triangles are proportioned to their opposite angles. That is, the 
greater the angle, the longer the opposite side, (meaning the side 
which does not touch the angle.) But this proportion is not direct, 
as 6 is to 18, so is 10 to 30, &c. But a table of sines, representing 
degrees, must be used instead of degrees. Students must be taught 
the use of the table of natural sines, by shewing only. 

NATURAL SINES. 

Sec. 39. It is not necessary to use tangents or secants in the ap- 
plication of trigonometry to the Mathematical Arts. Sines are suffi- 
cient in all cases. A table of natural sines is essential ; but artifi- 
cial sines and logarithms are not necessary. In some long series of 
calculations, particularly in working traverses of numerous courses, 
logarithms are a relief in the multiplications and divisions. Traverse 
tables, even then, are preferable. 

The student should always be fully instructed in trigonometry by 
the use of natural sines only. Afterwards, he will learn the use and 
application of tangents, secants, and logarithms, in two or three 
days, if required. 

Exhibition of Natural Sines. (See figure.) 

Take the triangle e s a. The angle near s is 30 degrees. The 
sine of 30° is a e. The co-sine of 30° is e x — which is equal to a 
s, and a s is considered and calculated as the co-sine of 30°. Now 
call the radius one inch, and the sine a e will be half an inch. Look 
in the table of natural sines, and you will see radius (sine of 90°) 
1.00000, and the sine of 30°, 0.50000, and the co-sine 0.86603. 
On measuring, all will be found thus to agree in measure with the 
table of sines, calling radius one. The same will hold true with 
every degree, minute, and second. 



30 



Directions for using the Abridged Table of Natural Sines. 

Sec. 40. The sine of every degree and of every half degree is 
set down in the abridged table. In all full tables the sine of every 
minute of each degree ° ' is included. For such tables the student 
is referred to the numerous books in use ; but they are too long for 
this treatise. A common scale may contain this abridged table ; 
and it will be found sufficiently accurate for all the common cases 
in practical Engineering, and in the Mathematical Arts in general. 
One minute of a degree will be the greatest error ; and in all cases 
below 45 degrees it will never deviate a minute. Consequently, in 
all traverse cases it is sufficient, with the aid of the square root ; and 
a useful substitute in all cases where full tables are not at hand. 

Sec. 41. As only the sines of all degrees and of all half degrees 
are given, the minutes of any half degree must be found, as follows : 
Take the marginal figure as the augment (or increasing number) 
for each minute in the degrees following it. This added to the last 
preceding given sine, will give the sine of the degree and minute re- 
quired. Thus : The sine of 14° 47 ' is required. The sine of 
14° 30' is set down as .25038 — the last sine preceding the sine re- 
quired. The marginal figure for the augment of all minutes of all 
the degrees from 10 to 17 inclusive, is 28. As 17' are to be added, 
28 times 17 must be added to .25038 (the sine of 14° 30'.) 28x 
17=476 — this number 28 (the augment for each minute,) being 
added, makes .25038+476=.25514. The true sine of 14° 47' is 
.25516 — but this error cannot throw the sine into either the minute 
above or the minute below. For tlie sine of 14° 48 ' would require 
35, and the sine of 14° 46' would be deficient by 26. If seconds 
of a minute are required, divide the augment by 60, and the quotient 
will be the addition to be made to the sine. 

Sec 42. When the middle term in the rule-of-three operation, is 
in sines, the answer will, of course, be in sines. To find the true 
minute to the sine thus obtained, proceed as follows : Subtract from 
the answer the sine of the degree, or half degree, next less than the 
answer. Divide the remainder, thus obtained, by the marginal 
figure (or augment) set against the degree. The quotient will be 



31 

the number of minutes to be added to the said given degree or half 
degree. 

ABRIDGED TABLE OF NATURAL SINES. 



3 ~ 


De- 


s. c-s 


c-s. s. 


De- 


3 T, 


M^ 


De- 


s. c-s. 


C-s. s. 


De- 


SB'S 


<s 


grees. 






grees. 


<i 


<| 


grees. 






grees. 


<| 


29 


0.00 0.00000 


1.00000 90.00 






23.00 


.39072 


.92050 


67.00,11 




.30 


.00373 


.99996 


.30 






.30 


..39875 


.91706 


.30 






1.00 


.01745 


.99985 


89.00 


i 




24.00 


.40 .74 


.913.55 


66.00 






.30 


.02618 


.99966 


.30 






.30 


.41469 


.90996 


.30 






2.00 


.03490 


.99939 


88.00 


1 


26 


25.00 


.42262 


.90631 


65.00 


12 




.30 


.04362 


.99905 


.30 






.30 


.43051 


.90259 


.30 






3.00 .0.5234 


.99863 


87.00 


1 




26.00 


.43637 


.69879 


64.00 






.30 


.06105 


.99813 


.30 






.30 


.44620 


.89493 


.30 






4.00 


.08976 


.997.56 


86.00 






27.00 


.45399 


.89101 63.00 


13 




.30 


.07846 


.99692 


.30 






.30 


.46175 


.88701 


.30 






5.00 


.08716 


.99619 


85.00 


2 




28.00 


.46947 


.88297 


62.00 






.30 


.09585 


.99.540 


.30 






.30 


.47716 


.87882 


.30 






6.00 


.104.53 


.99452|84.00 






29.00 


.48481 


.87462;61.00!l4 




.30 


.11320 


.99357 .30 






.30 


.49242 


.870.36 


.30 






7.00 


.12187 


.99255 83.00 


3 


25 


30.00 


.50000 


.86603 


60.00 






.30 


.130.53 


.99144 .30 






.30 


.50754 


.86163 


.30 






8.00 


.13917 


.99027 82.00 






31.00 


.51504 


.85717 


59.00 


m 




.30 


.14781 


.98902 .30 






.30 


..522.50 


.8.5264 


.30 






9.00 


.15643 


.98769 


81.00 


4 




32.00 


..52992 


.84895 


58.00 






.30 


.16505 


.98629 


.30 






.30 


.53730 


.84339 


.30 




28 


10.00 


.17365 


.98481 


80.00 






33.00 


.54464 


.83867 


57.00 


loi 




.30 


.18224 


.96325 


.30 






.30 


.55194 


.83389 


.30 






11.00 


.19081 


.96163 


79.00 


5 


24 


34.00 


.55919 


.82904 


56.00 






.30 


.19937 


.97992 


.30 






.30 


.56841 


.82413 


.30 






12.00 


.20791 


.97815 


78.00 






35.00 


.57358 


.81915 


55.00 


16i 




.30 


.21644 


.97630 


.30 






.30 


.58070 


.81412 


.30 






13.00 


.22495 


.97437 


77.00 


6 




36.00 


.58779 


.80902 


54.00 






.30 


.23345 


.97237 


.30 






.30 


.59462 


.80386 


.30 






14.00 


.24192 


.97030176.00 




23 


37.00 


.601-2 


.79864 


53.00 


17 




.30 


.25038 


.96815' .301 




.30 


.60876 


.79335 


.30 






15.00 


.25882 


.96.593',75.001 7 




38.00 


.61.566 


.78801 


52.00 






.30 


.26724 


.96363 .30 






.30 


.62251 


.78261 


.30 






16.00 


.27564 


.96126 74.00 






39.00 


.62932 


.77715 


51.00 


18 




..30 


.26402 


.9.5882 .30 






..30 


.63608 


.77162 


.30 






17.00 


.29237 


.9.5630 73.00 


8 


22 


40.00 


.64279 


.76604.50.00 




1 .30 


.30071 


.9.5372 .30 






.30 


.64945 


.76041 


.30 




27 


18.00 


.30902 


.95106 72.00 






41.00 


.6.5606 


.75471 


49.00 


19 




.30 


.31730 


.94832 .30 






.30 


.66262 


.74896 


.30 






19.00 


.32.5.57 


.94.5.52 71.00 


9 


21 


42.00 


.66913 


.74314 


48.00 






.30 


.33381 


.94264 


.30 






.30 


.67.5.59 


.73728 


.30 






20.00 


.34202 


.93969 


70.00 






43.00 


.68200 


.73135 


47.00 


20 




.30 


.3.5021 


.93667 


.30 






.30 


.68835 


.72537 


.30 






21.00 


.35837 


.93.358 


69.00 


10 


20i 


44.00 


.69466 


.71934 


46.00 






.30 


.36650 


.93042 


.30! 




.30 


.70091 


.71325 


.30 






22.00 


.37461 


.92718 


68.00 




45.00 


.70711 


.70711 


45.00 






.30 


.38268 


.92383 


.30 






.30 


.71325 


.70091 


.30 





32 



Illustrations of Trigonometey. 

The different modes of applying the Square Root and the Rule 
of Three, to cases of Trigonometry, are five.* 

Sec. 43. Any two sides of a right angled triangle being given, ilie 
other side may he found hy the square root, ivithout the aid of angles, 

1. Illustration. If the height of a wall is known to be 20 feet, 
and a ditch at the bottom of the wall is 12 feet wide; the length of 
a ladder for scaling the wall may be found, by squaring the height 
of the wall and breadth of the ditch separately, adding the two pro- 
ducts and extracting the root of the sum : 



20 


12 


400 


544.0000(23.32 Answer. 


20 


12 


144 


4 


too 


144 


544 


43)144 
129 

463)1500 
1389 

4662)11100 
9324 



2. Illustration. If the width of the ditch and length of the lad- 
der are known, the height of the wall may be found by squaring 
the length of the ladder and the breadth of the ditch separately ; 
then subtracting the square of the breadth of the ditch from the 



* In all cases, a student ought to plot a triangle, before commencing the calculation. An 
old experienced engineer can obviate numerous perplexities, by making hasty random 
sketches of whatever figures be may attempt to calculate. 



33 

square of the length of the ladder, and extractbg the root of the re- 
mainder. 



23.32 


12 


543.8224 sq. ladder. 


23.32 


12 


144 sq. ditch. 


4664 


144 


399.8224(19.99 Answer. 


6996 




1 


6996 






4664 




29)299 






261 


543.8224 




389)3882 
3501 

3989)38124 
35901 



2223 

Loss by omitted fractions reduced 20 feet, one hundredth of a 
foot. 

Sec. 44. One side of any triangle, and two of the angles, being 
given; the other angle may be found by subtraction, and the other 
two sides by the rule of three with sines of the angles. 

1. Illustration. The distance between a corner of the school, 
house and the steeple of the church is required. An imaginary 
triangle is so made, that the steeple is in one angle, the comer of 
the school-house in another, and a stake is set up in the yard, 80 
feet from the corner of the school-house, for the otiier. The com- 
pass, or quadrant, is set at the corner of the school-house; by 
which it is found, that the imaginary lines to the church and stake 
form an angle of 80 degrees at this corner. The compass is then 
set at the stake ; by which it is found, that the imaginary lines to ' 
the church and school-house, form an angle of 70 degrees at the 
stake. As every triangle contains 180 degrees, on subtracting the 
80 and the 70 degrees (that is, 150) from 180, it will leave the an- 
gle at the church 30 degrees. We shall thus have obtained one 
side of a triangle 80 feet, and all the angles, 80, 70, and 30. We 

5 



34 

then state— if 30 degrees (at the church) give 80 feet, (from the 
school-house to the stake) what will 70 degrees (at the stake) give ? 
But, as sines of degrees must be used instead of degrees, the opera- 
tion is as follows : 

As the sine of 30° (at the church) is to 80 feet, so is the sine of 
70* (at the stake) to the distance from the school-house to the church. 

30" : 80 :: 70° 
Sine .50000 Sine .93969 

80 



[brought up.] 417520 )84.17520(168.3 Answer. 

400000 50000 



175200 341752 

150000 300000 



Remainder 25200 41752 

We find the distance between the school-house and church to be 
168 feet and three-tenths of a foot, with an immaterial remainder. 

The distance between the stake and the church may be found by 
the same process, taking the sine of the angle at the school-house 
for the first number. 

Note. Future examples and descriptions will be less minute on 
the application of the rule -of -three and sines to Trigonometry ; there- 
fore the student must work out numerous examples (furnished by 
the teachers, or from other books,) under this mode of application. 

2. Illustration. The distance from the centre of the earth to the 
moon is required. Imagine a triangle, of which the moon is at one 
angle, the centre of the earth at another, and a place where you 
stand is at another. The angle where you stand is 90 degrees ; 
because the sensible horizon is one side of the triangle, and a per- 
pendicular Hne to the centre of the earth is another. The angle at 
the moon is 57J- minutes of a degree (as found by the parallax, 
hereafter to be demonstrated.) We then state : if 57J- minutes of a 
degree give 4000 miles (the distance, near enough, to the centre of 
the earth,) what will 90 degrees give ? Thus : 



35 

571 : 4000 m. :: 90^ 
Sine .01663 1.00000 1.00000 Sine. 



.01663)4000.00000(240,529 Answer. 
3326 



6740 
6652 

8800 
8315 



4850 
3326 

15240 
14969 

The distance of the moon from the centre of the earth, is gene- 
rally set down at 240.000 m. I have always made it a little 
more. 

Sec. 45. If two sides of any triangle are given, and one of the 
angles which is opposite to one of the given sides ; the other two angles 
and the other side, may le found hy the rule of three and sines. 

Illustration, We are on the east hill, and wish to ascertain the 
length of a level air-line to the west hill, at an equal height. The 
two adjoining sides are straight, smooth, and convenient for chain- 
ing ; and meet at a well defined angle at their bases. The east 
side hill has a slope of 196 feet — the west side hill has a slope of 
360 feet — the angle made on the east hill by its slope and the hori- 
zontal level is 72°. We state as follows : 



36 



As 360 feet : to 72° : : 196 feet to the angle on the west hilL 



360 f. 



: 72° 

Sine .95106 
196 



196 f. 



.51777=31° 11' 



570636 
855954 
95106 



)186.4076(.51777=31° 11' 
1800 



[carried up.] 186.40776 



640 

360 

2800 
2520 

2807 
2520 



The angle on the east hill, 
The angle on the west hill, 



2876 
2520 

72° 

31° 11' 



The two angles equal 103° 11' 
Subtract from 180° 

The other two angles 103° 11' 



Angle at the base, 76° 49 ' 

Find the air-line thus : 
72° : 360 :: 76° 49' 
.95106 .97365 

360 



5841900 
292095 

[carried up.] 350.51400 



350.51400(379 f. Answer. 
275318 



751960 
6658.42 

862180 
855954 



6226 Rem. 



37 

Sec. 46. If two sides of any triangle are given, and an angle form- 
ed at their meeting ; the other two angles and the other side may ie 
found iy the rule of three, sines, and the square root. 

Illustration. A road was traversed with chain and compass, 
through an uneven parish. At one place the turn was so short, that 
it was necessary to strilce a circle to cut through three of the cor- 
ners (being at the meeting, and ends, of two of the surveyed lines.) 
They run thus : north 44 degrees east 8 chains 50 links ; and north 
60 degrees east 12 chains 40 links. By drawing a north-and-south 
line through the place of their meeting on the plot, and reversing 
the first line (calling it south 44 west) and subtracting the 60 degrees 
of the second line, from 180, the meeting angle will be found to be 
164°. Imagine the first line extended beyond the angle so far, that 
a perpendicular, let fall from the end of the second line, will strike its 
termination. Then the first line, ideally extended, and the imagin- 
ary perpendicular, will form the two legs of a right-angled triangle. 
The side sought for, connecting the extreme ends of the two surveyed 
lines (which is to be the chord line of the proposed arc,) will be the 
hypothenuse ; and is found by the square root, in the usual way. 
As these two ideal lines and the second known side, form a right- 
angled triangle, they are found in the usual way, by the rule of three 
and sines — the 164° being subtracted from 180° ; leaving the ad- 
joining angle (within the new triangle) 16°. Thus : 

180° 90° 

164° 16° 

16° one acute angle. 74° the other acute angle. 

90° : 12.40 : : 16° 

1.00000 .27564 

12.40 



pendicular in chains, links, and decimals. 



1102560 
55028 
27564 

3.3179360 the per- 



90° : 


12.40 


:: 74° 


1.00000 




.96126 
12.40 




3845040 






192252 






96126 



11.9196240 the hori- 
zontal leg in chains, links, and decimals. 

Add the horizontal leg obtained, to the first surveyed side, as 8.50 
-[-11.92=20.42 for the whole horizontal leg. Then find the chord 
line of the proposed arc thus : 

20.42 3.32 

20.42 3.32 



4084 664 

• 8168 996 

40840 996 



416.9764 11.0224 

11.0224 



427.99 88(20.64-1-Chord line of the proposed arc ; 
4 which arc is to strike the meeting 

- angle and the extreme ends of 

406)2799 the two surveyed lines. 

2436 



8124)36388 
32496 

Note. This is a very important application of trigonometry in 
laying out rail-roads, McAdam roads, canals, &c., as will be shown 
farther on. The operation is often performed with a table of tan- 
gents ; but I can perceive no advantage in driving the practical 
mathematician from the table of natural sines. 



39 

Sec. 47. If one leg of a right angled triangle is given, and the 
sum of the hypothenuse and the other leg, the tohole triangle may he 
completed iy the square root, the rule of three, and sines. 

Illustration. A green-house (for the defence of plants which 
cannot brave our wmters, but do not need the hot-house) was 
threatened by the fall of a decaying poplar, 22 feet south of its 
sash-glass roof, which extended to the ground. The top of the tree 
had been previously cropt, at the height of 64 feet. It appeared to 
be necessary, that the trunk should be cut, so as to fall at the exact 
south margin of the green-house. The proprietor of the green- 
house called on a neighboring practical mathematician, to ascertain 
how high from the ground this tree must be cut, so that (hanging 
by the bark and sap-wood as by a hinge) it should fall with its top 
just at the outer edge of the green-house sashes. His inquiries 
were answered as follows : Square the given base (22 feet,) and 
square the height of the tree (64 feet ;) add the two squares, and 
extract the root, which will give the hypothenuse from the top of 
the tree to the margin of the sashes. 

Consider this hypothenuse as one side of a right angled triangle ; 
and find the angle at its top as in other cases of right angled trian- 
gles, where all the sides are given — it will be 18° 58'. Imagine the 
tree to be cut as intended, and the top fallen. As the supposed 
fallen part and the same part erect, are equal, the angles at the top 
and bottom are equal — that is, each is 18° 58'. Add these two an- 
gles together and subtract their sum from 180, which will give the 
middle angle of 142° 4'. Subtract the middle angle from 180, 
which will give the top angle of the triangle sought, 37° 56'. Then 
proceed as in all cases of a right angled triangle, with one side and 
one angle given. It should be remarked, that the top angle of the 
created or borrowed triangle, is not that of the sought triangle ; 
consequently, that double the top angle produces the top angle of 
the sought triangle. 

Other methods are given in books ; but this is the most simple. 
This case is very important in conic sections ; particularly the ellipse. 



40 

Sec. 48. The three sides of any triangle being given, it may he 
completed, also its area found, hy the square root, rule of three, and 
sines. 

Illustration. One rafter of an unequal roof is 40 feet, the other 
20 — the cross-beam is 50 feet. This oblique-angled triangle must 
be divided into two right angled triangles. This will require that 
a perpendicular be let fall from the meeting of the rafters, to the 
beam, so as to divide it into two parts, each according to the length of 
the rafter stretched over it, thus : 

As the length of the beam 50 feet, is to the sum of the lengths of 
the two rafters (40+20) 60 feet, so is the difference of the two raf- 
ters (40 — 20) 20 feet, to the difference between the parts of the 
beam into which the perpendicular divides it. 

20 2)24(12 the half difference. 



2)50(25 half the beam. 
12 

13 shortest half of the 
beam. 





)0 




: 60 








20 


)50 






1200(24 


— 






100 


25 


h. 


b. 




12 






200 
200 



37 longest half of the beam. 



Having obtained the base leg of each of the two right angled 
triangles; and the hypothenuse (rafters) of each being given, the 
perpendicular leg to both, and the acute angles of both, may be 
found as in all cases where the hypothenuse and one leg are given. 

This rule may be applied in finding angles in fields, which had 
been measured without a compass or other instrument for taking- 
degrees. 

Sec 49. Of the three angles and three sides of every triangle, 
three of these six constituents must be known, and one of the known 
constituents must be a side, excepting the corner w^here the square 
root applies. But in a right-angled triangle and in an isosceles 
triangle, one side and one angle only are to be taken in the field. 
Because every right-angled triangle contains one right angle, and 
every isosceles has two equal-angles. The isosceles triangle is of 
ffreat use in rail-road curves and other curvilinear calculations. 



41 

Sec. 50. In figures of many sides, it is often convenient to know 
at sight, tlie sum of the degrees contained in all the angles, outer 
and inner. Count the angles — deduct two from the whole — multiply 
the remainder by 180. This is too evident from inspection to re- 
quire illustration. Take a figure of 7 sides : 7 — 2=5x180=900 
degrees. 

Sec. 51. Examples in all the cases ; hut they are not set down in 
systematic order. 

1. The height of a tree is required. The distance to the tree is 
30 feet, the angle made by a line to the top of the ti'ee, with a line to 
the bottom, is 30° 15' — the ground to the tree is level, and the tree 
is perpendicular. 

2. A rope to the top of a tree is 62 feet long ; the angle made by 
the rope and a line to the bottom of the tree, is 42° — the ground is 
level, and the ti'ee perpendicular. How high is the tree ? 

3. Two stakes on the east side of the river, are 2 chains 64 links 
apart. The angle at the north stake, formed by the measured line 
between the stakes, and an imaginary line drawn to a cedar tree on 
the west side of the river, is 79° — the angle at the south stake, 
formed by said measured line, and an imaginary line drawn to said 
cedar tree, is 83°. How far is it across the river, from the south 
stake to said cedar? 

4. A dam is 900 feet long. An imaginary line, drawn from a 
pine tree on the west shore of the river, to the east end of the dam, 
forms an angle of 68° with the line of the dam. How far is the pine 
below the west end of the dam, the shore being perpendicular to it? 

5. Determine whether the outer base-sills of a house form per- 
fectly square corners. Sills of the sides, 48 feet — of the ends, 37 
feet. Find the diagonal by the square root. Then measure with 
a tape, and see whether the measured diagonal agrees with the cal- 
culation. If not, rack the whole base with a lever until it will agree. 

6. The upper ceiling of a room is 11 feet 6 inches high, the 
floor is 32 feet 9 inches long. Find the length of a tape required, 
to reach from the upper ceiling at one end of the room, to the floor 
at the other end, by the square root. 

7. The length of a brace is required from shoulder to shoulder 
inside, and from shoulder to shoulder outside — the centre of the mor- 



42 

tice, on the beam, is 7 feet from inner meeting of the beam with the 
post; and, on the post, 9 feet. Determine the length by the square 
root. 

Note. The angle of the mitred shoulders of the braces are de- 
termined by a reference to the homologous sides between the shoul- 
ders and the right angled triangle, formed by the brace, post, and 
beam. 

8. You take a station where you can see a flag on each side of 
the base of a mountain ; you wish to ascertain the distance through 
the mountain from flag to flag. Find the distance from your station 
to both flags, by trigonometry or measure — also take the angle at 
the station. Then find the distance through the mountain, as in all 
other cases, where two sides and a contained angle are given, for 
finding the other side. 

9. A straight line to be drawn obliquely through a block of 
houses is required. You can contrive to compass the block by a 
base line 140 feet long ; and two other lines meeting at the point 
where the measure is to fall perpendicularly on the base, one 92 
feet and the other 67 feet ; but could take no courses, nor angles. 

10. You purchased a piece of ground, which is to be in the form 
of a right angled triangle, with the base leg 20 chains. The sum 
of the perpenciicular leg and the hypothenuse to be 60 chains. 
What will be the length of the perpendicular leg, what will be the 
length of the hypothenuse, and how much land will be included in 
the triangle ? 

MENSURATION* 

IS DIVIDED INTO SUPERFICIES AND SOLIDS, 

Sec. 52. Superficial Mensuration. 

1. Parallelogram. A garden is 50 feet wide and 200 feet 
long. All the sides meet at right angles. It is manifest a strip 
from one end, that is one foot wide, contains 50 feet, another foot 
takes in 50 feet more, and so on. Therefore the whole 200 feet 

* As Mensuration, when practically applied, requires more or less extended illustration, 
according to the peculiar circumstances of cases, I shall merely give the most common rules 
in use in few words— reserving for special applications such further demonstrations as mas- 
appear necessary. 



43 

contains 200 times the first 50 feet strip. Hence the rule : multiply 
the length and breadth together and the product will le the square 
feet contained in a parallelogram or square. 

2. Tkiangle. If a diagonal line is drawn through the afore- 
said garden, it will be divided into two equal parts. Consequently- 
each part will be a triangle containing half as mucli as the whole 
garden. As the long side of each triangle, called the base, is 200 
feet, and the short side, called the perpendicular, is 50 feet, it fol- 
lows that if the base is multiplied by the perpendicular, the pro- 
duct will be double the true contents. Hence the rule : multiply 
the lase of a triangle hj its perpendicular, and halve the product; 
which will give the square feet contained in it. 

3. Regular polygon. Take for example an octagon. It may- 
be considered as consisting of eight isosceles triangles, whose apex- 
es meet at the centre and whose bases constitute the eight sides. 
Now if all these eight bases are added together and that sum mul- 
tiplied by one perpendicular, it is manifest that the product will be 
double the contents of all the eight triangles. Hence the rule: 
multiply the sum of all the sides of a regular polygon iy their dis- 
tance from the centre, and half the product will he the area. 

4. Circle. Suppose the sides of the aforesaid polygon to be 
indefinitely short, so as to form a polygon of many million sides ; 
it is manifest that tlie arc of a polygon, called a circle, may be 
found as of polygons with longer sides. Hence the rule : multiply 
the periphery of a circle ly half its diameter, and half the product 
will he the area. 

5. Periphery of a circle is produced by multiplying the diameter 
by 3.1416 — ^more accurately, 3.14159. 

6. DiABiETER OF A CIRCLE is produced by dividing the periphery 
by 3.1416. 

7. Area of a circle may be produced by squaring its diameter ; 
and then displacing the corners by multiplying the product by .7854. 
For example, suppose the diameter of a circular garden bed 10 feet. 
Multiplied by itself the product is 100. Multiply the 100 feet by 
.7854, and the true product will be 78.5400 or — 78 feet 54 hun- 
dredths. 

8. Length of any arc of a circle may be found by multiplying 
the degrees of the arc, the radius, and the formula .01745, into each 



44 

other. Thus 40 degrees of an arc with a radius of ten inches will 
stand thus : 40xl0x.0l745=6.98 inches. 

9. Sector of a circle may be found by multiplying the length of 
the arc (as found above) by the length of the radius, and halving the 
product. This is an application of the same principle which is ap? 
plied to the regular polygon and circle. 

10. Segment of a circle may be found by first finding the area 
of the sector and then subtracting the area of the triangle made with 
the chord line and two radii. But this triangle must be added, if 
the segment is greater than half the circle. 

11. Area of an oval may be found as the area of a circle. 
That is, by bringing the oval to a circle by multiplying the longest 
diameter by the shortest, and that product by .7854. 

12. The superficies OF a prism or cylinder may be found by 
multiplying the pyrimeter of either end by the length, and adding the 
area of both ends. 

13. The superficies of a pyramid or cone may be found by 
multiplying the pyrimeter of the base by the slanting side, and halv- 
ing the product — to this add the area of the base. 

14. The area of a parabola may be found by multiplying the 
base by the perpendicular and deducting one third of the product. 

15. The superficies of a globe may be found by multiplying 
its circumference by its diameter. If the superficies of a segment 
is required, multiply the whole circumference by the height of the 
segment. 

Sec 53. Solid Mensuration. 

1. The solid contents of a cube, parallelopiped, prism, or 
cylinder, may be found by multiplying the area of one end (or 
side) by the length. 

N. B. A wedge may be considered as a triangular prism ; the 
edge forming one angle of the prism, and the corners of the head of 
the wedge, as the other two angles of the prism. 

2. The solid contents of a globe may be found by multiplying 
the siH'face by the diameter, and taking one sixth of the product for 
the solid contents. 



45 

3. The solid contents of a pyramid or cone may be found by 
multiplying the area of the base by the perpendicular, and taking 
one third of the product for the solid contents. 

4. The height of a pyramid or cone may be found by the rule 
of proportion, if the height and upper and lower diameters of any 
frustum of it is given, thus : as the difference between the diameters 
is to the height of the frustum, so is the upper diameter to tlie height 
of the part above the frustum. If this be added to the height of the 
frustum, the sum will be the height of the whole pyramid. 

5. The solid contents of the frustum of a pyramid may be 
found by finding the height of the part above the frustum, as direct- 
ed in the last rule ; then finding the solid contents of the whole pyra- 
mid, and of the part above the frustum, as before directed, and sub- 
tracting the latter from the former — the remainder will be the solid 
contents of the frustum. The same rule applies to the cone ; but 
after the above process is finished, the said remainder must be mul- 
tiplied by the formula, .7854. Example : Lower diameter of the 
given frustum is 10 inches, upper 6, height 8. As dif. 4 : 8 : : 6 : 
12. Height8 and 12=20. 10x10=100x20=2000. 6x6=36 
X12=432. 2000—432=1568 divided by 3=522.66 Answer. 
Contents of pyramid multiplied by .7854=^410.49 Answer. 

6. The solid contents of the frustum of a pyramid* may be 
found by multiplying the breadth of the top by the breadth of the 
bottom, and multiplying that product by the height. To the last 
product add a sum, produced by squaring the difference between the 
breadth of the top and bottom, and multiplying that square by one 
third of the height. 

If it is the frustum of a cone, the last product alone must be multi- 
plied by the formula, .7854. 

7. Guaging. The parts of a cask on each side of the bung are 
frustums of cones, as A B, A B, (see figure) are frustums of A C, 
A C, considering the staves as straight from the bung to the heads; 
therefore their contents are found in the same way, separately. 
By doubling them, the contents of both frustums, or the whole cask, 
is found. 

* I Imve never seen this method published. As far as I know, it was first used at the 
Rensselaer Institute. It may be demonstrated by a model. 



46 

But the staives generally curve more or less ; for which an allow- 
ance must be made, unless the convexity of the inner surface of the 
heads occupies a space equal to what is added by such curviture. 
Common casks require that about a tenth part of the difference be- 
tween the head diameter B and the bung diameter A, be added to 
the bung diameter A. This will increase the contents so as to 
equalize the contents of the curvitures e e e e. 

For expeditious practice in guaging, graduated guaging rods are 
used. Or the following rule and formulas may be adopted. 

8. Add the head and bung diabietees A and B (taken in inch- 
es) and take half that sum for the average diameter. To this ave- 
rage diameter add a sum equal to one eighth of the difference be- 
tween the two diameters, for the curviture of the staves e e e e in 
common casks. Square the last sum and multiply it by the length 
of the cask B B. As this gives too many square inches, divide the 
product as follows: if for Avine of 231 inches to the gallon, divide 
by 294.12 — if for beer, cider, ale, &c., of 282 inches to the gallon, 
divide by 359.05. The quotient in both cases will be in gallons. 
If for bushels of 2150.4 inches to the bushel, divide by 2738. 

9. The tonnage of a ship, according to a statute of the United 
States, must be found as follows : Take the length of the vessel 
from the fore part of the main stem to the after part of the stern 
post above the deck or decks ; and the breadth at the broadest 
part above the main wales, and deduct from the length three fifths' 
of the breadth. 

For a douiJe-decked vessel, half of the breadth shall be account- 
ed the depth; then the length, breadth and depth must be multipli- 
ed together, and the product divided by 95 — the quotient will be the 
tonnage. 

For a single decked vessel, the depth must be taken from the un- 
der side of the deck plank to the ceiling in the hold. In all other 
respects proceed as with double deckers. 

Ship carpenters measure is made by proceeding as above in all 
respects, excepting that they take the length of the keel, the breadth 
of the main beam, and the depth of the hold ; though in double- 
deckers they take half of the breadth for the depth as before stated, 
under douUe-decked vessels. 



47 
LAND SURVEYING, (Geodesia,) 

IS DIVIDED INTO FOUR KINDS. 

Sec. 54. (PedioJiietry.) Field surveying. This is applied by 
farmers, when the contents of a Jield are required for the purpose 
of ascertaining its productiveness by the acre, for determining the 
quantity of seed to be sown, or to estimate the value of labor in 
ploughing, mowing, harvesting, &c. 

Sec. 55. {Agrometry.) Farm surveying. This is applied 
when the out-bounds, courses, distances, and contents, of a farm or 
lot are required, for the purpose of description in a deed, for the es- 
tablishment of boundaries, for ascertaining the contents, for making 
a map, &c. For this kind of survey the courses and distances are 
not given. 

Sec. 56. (Oromefry.) Line surveying. This is applied when 
the obscure or lost lines of a previous survey are to be revived and 
marked ; or where lines are given, by a mere plot upon paper, to 
be traced and marked. For this kind of survey, the courses and 
distances are given, which are taken from the previous survey, or 
from a measurement made with instruments on the paper plot. 

Sec. 57. (Udrometry.) Aquatic surveying. This is applied to 
harbors, riv#rs, lakes, ponds, marshes, &c., where the measure of 
surfaces, the location of shoals, depths of water, quality of the under- 
laying earth, &c., are required, in places which cannot be approach- 
ed on dry land in the usual way. 

Sec 58. These four names are compounds of the Greek metreo 
(to measure) with the following Greek words : 1. Pedion (neuter) a 
plain, open, level field. 2. Agros (masculine) ground, land, a farm. 
3. Oros (mascuhne) a boundary, a land-mark. 4. Udor (neuter) 
water. These derivations are given to aid the memory of the stu- 
dent ; while the names will assist him in systematizing his views of 
practice. 

1. PEDIOMETRY. {Field surveying.) 

Sec. 59. As any figure bounded by straight lines, can be cut into 
triangles, and as the contents of any triangle may be found by mul- 
tiplying its base by a perpendicular line, drawn in the nearest direc- 



tion from it to the opposite angle ; it follows, that the contents of any 
open field may be found by cutting it into triangles. 

Sec. 60. Use of the cross. In the figure, suppose the dotted lines- 
to represent imaginary lines in the field, actually measured thus. 
Measure with a rope or chain from corner to corner, as from B to 
H — from H to G — and from D to F. After measuring these lines, 
take the perpendicular, with a cross. The cross may be made by 
laying two laths or strips of board across each other, and nailing 
their centre to the top of a staff. With the cross move back and 
forward, until the point is found, where one shp of the cross points 
to the angle, as at A, while the other lines point at the two ends of 
the base line, as at B and H, then measure the perpendicular. In 
this way take the base and perpendicular of every triangle, and mul- 
tiply each base by its perpendicular. Add all the products, and 
halve that sum ; which will give the contents of the field in the 
squares of whatever measure was used — if the rope was in feet, the 
answer will be in square feet ; if in rods, in square rods ; if in chains, 
in square chains. 

Sec. 61. Field calculation. If the number of acres are required, 
and the measure was taken in chains and links, the calculation is 
very simple. Example. Base B H 12.30, perpendicular to A 
7.21, and perpendicular to C 8.42. These two perpeiidiculars may 
be added together before multiplying ; because they have the same 
base. Then 7.21 + 8.42=15.63x12.30=192.24. Base H G 
11.05 per. to C 4.00 multiplied together =44.20. Base D F 11.96 
per. to C 10.21 and per. to E 7.15 ; which perpendiculars added 
make 17.36 multiphed by 11.96=207.62. Add the products of all 
the triangles 192.24+44.20+207.62=444.06. Half of that sum, 
222.03 is the contents of the field in square chains and links. As 
ten square chains are an acre, divide the chains by 10, which gives 
22 acres. Multiply 2.03 by 4 and divide by 10, which gives 32 
rods. The answer is 22 A. Q. 32 R. omitting all fractions below 
links. But if there are numerous triangles and accuracy is required, 
the fractions must be retained until the operation is finished, and 
merely rejected from the final answer. 

Sec. 62. Compass necessary. The student will perceive, that 
maps cannot be so drawn from such surveys as to give the points 
of compass; neither can deeds be drawn by them. The next 



49 

method must be resorted to in all such cases ; also in fields where 
hills, woods, &c., obstruct the sight. 

II. AGROMETRY. {Farm surveying.) 

Sec. 63. Remark. This being the most important kind of sur- 
veying, and that to which all the other kinds will be referred for 
their chief explanations ; I shall give directions in a most familiar 
manner. I can devise no plan more eligible, than that of arrang- 
ing the directions under distinct heads ; so that the whole shall 
form the history of a survey. 

History of an Agrometric Survey. 

Sec. 64. Having received an application to survey a farm own- 
ed by two brothers ; and to divide the same, unequally, as to quan- 
tity, between them, I proceeded as hereafter related. This survey 
is a real case in iny practice ; excepting that L took in parts of 
three other real surveys, for the sake of making it more diversified. 

Sec. 65. Magnetizing the needle. With a view to execute the 
job with accuracy, with a strong magnet I retouched the needle of 
my compass. As magnets communicate contrary poles and attract 
contraries, the only safe method of touching the needle is, to bring 
it near the magnet, suspended on a pin's point as a pivot, and let its 
poles choose for themselves. Then rub the pole of the needle on 
the end of the magnet it has chosen ; and the other pole on the 
other end. 

Sec. 66. Correcting chain. I remeasured my chain, and added 
a few wire rings to bring it to the precise measure of 66 feet ; and 
also to equalize all the links — carefully counting the hundred links, 
and equalizing the quarters (or twenty-fives) into rods, and seeing 
that the whole chain was accurately tallied off in tens of links. 

Sec. 67. Stakes and tallies. I counted over my nine wire stakes, 
and supplied the deficiencies; having each 12 inches long, with a 
well-turned eye at the top. In each eye I tied strips of red, white, 
and black rags; that they might be seen readily among dead 
leaves, evergreens, &c. I made a new set of 7 leather taUies ; 
which consisted of inch-square pieces of sole-leather, strung upon a 

7 



50 

cord passed through their centres. The cord was of a sufficient 
length to pass around the waist of the hind chain-bearer. 

Sec. 68. Field hook, pens and ink. I prepared a blank book, 
four inches square, covered with thin leather ; fitted to carry in the 
left breast-pocket. And I sewed a slender vial in the fore edge of 
the same pocket, containing good ink, absorbed by cotton wadding. 
One short pen was set into the vial ; and several other short quills 
were fastened in the bottom of the same pocket. 

Sec. 69. Scale, dividers, protractor and ruler. Next I screwed 
up the joint of ray best dividers, and made the points sharp and 
smooth as the points of sewing needles. Selected my best ivory 
scale, and most accurate semi-circular protractor — also, a wooden 
ruler, one inch wide, with its opposite edges precisely parallel, 
straight, and thin ; being supported in a perfectly straight form by 
a high ridge along the centre of the upper side. This ruler is for 
drawing random parallel meridians, by its opposite edges. 

Sec. 70. A protractor for laying parallels. My protractor being 
of the common form, I ground down the upper side of the straight 
part to a sloping bevel, and engraved a line along the face of the 
bevel, near its edge, and exactly parallel to it. This line is for 
setting in a foot of the dividers, when laying the straight side of the 
protractor parallel to a meridian line. 

Sec. 71. Taking elevations and depressions, and moderate heights. 
My employers did not belong to that penurious class, who prefer a 
hurried, half taken survey, to expending another dollar to gain fifty 
in utility. Therefore, I prepared for taking all the inequalities of 
surface along the line, worthy of notice ; also the heights and dis- 
tances of important buildings, &c. A very simple and cheap in- 
strument is Kendall's tangent-scale. This is prepared by striking a 
quadrant with a radius equal in length to the distance between the 
outsides of the sights. Make a tangent to this quadrant, and set off 
the degrees and half-degrees upon a scale ; rather upon the edges of 
the sights, to be continued on a scale when required. By ranging 
upwards from the bottom of the slit in the opposite sight, or ranging 



51 

downwards from the tangent scale, through the same, ascents and 
descents may be taken with sufficient accuracy.* 

I put up my brass slip for reducing the farm to a Single triangle, 
which will be described in its proper place, with its application. 

Sec. 72. Assistants. On the morning appointed, I repaired to 
the farm to be surveyed. !• found all the assistants in readiness as 
I had directed. One was selected for hind chain-hearer ; because 
he had more talents and learning than the rest. On the accuracy 
of the hind bearer depends the correctness of the chaining. He 
directs the fore bearer so as to keep him in range with the flag, he 
receives all the nine stakes, wears the string of leather tallies, and 
slides one for every ten chains, and renders the account of chains 
and links when called on by the surveyor. His pay was one dollar 
and twenty-five cents per day. The fore chain-bearer — pay fifty 
cents per day. Ax-man — pay fifty cents per day. Baggage-man 
— pay fifty cents per day. Flag-man — pay seventy-five cents per 
day; for his duty, though very light, requires considerable judg- 
ment and great care. 

Sec. 73. The first step taken at the farm waste go around it, and 
put up a stake at every place where the line turned sufficiently to 
change the course. I had advised my employers to have their in- 
terested neighbors present, to see the boundaries fixed, to avoid fu- 
ture controversy. They all attended us and all the men employed, 
except the baggage-man. Him I directed to cut a perfectly straight 
flag-staff, 10 feet long, and mark it off" into feet, and wind spirally a 
red, white, and black cloth, around three feet of its upper end. And 
advised my employers to have put up in a pack or basket,f such 
articles of refreshment as we might need, in order to save the time, 
which would be consumed while returning to take formal meals. 

* This simple operation was suggested to me in the year 1831, by Mr. Thomas Kendall of 
New Lebanon Its utility has been tested sufficiently for tive years at Rensselear Institute. 
Hanks' compasses are furnished with an appendage, well adapted to this, and other useful 
purposes. Aleneely has revived the method which was approved and in common use about 
the midiUe of last century. But it is the cheapness and univeisal application, that gives 
Kendalls method its value. 

t In surveys of new lands, where tlie party sleep in woods, the baggage-man must carry 
an oil-cloth to spread under, and a tent to stretch ovtr, the whole company. A poney, with 
a leather cover sewed to the fore end of the saddle and spreading back over all the baggage, 
to defend it from storms and from being scratched and torn, is useful. 



52 

All the other men went around with us to cut and set up stakes, to 
throw stones about the principal corners ; the flag-man was directed 
to take particular notice of the corners, so that he might readily 
find them. 

Sec. 74. Fixing first corner. Having set up stakes at all the 
corners, we entered upon the survey. As the employers were to 
exchange deeds of release, which might lay the foundation for fu- 
ture subdivisions among heirs and purchasers, as well as between 
the present owners, I was careful to fix the boundary for the place 
of beginning by reference to objects which might be found in fu- 
ture, as follows : 

Beginning at a stake and stones, standing on the east bank of 
Stoney Brook, at a place thirteen chains from the junction of said 
brook with Meadow Brook on a course N. 10 W. and eleven chains 
below Griswold's mill, on a course, along said brook, S. 23 W. 

Sec. 75. Taking the course. I set my compass-staff into the 
ground at the said stake, marked 1 in the map. Having sent the 
flag-man to the corner marked 2, I directed the sight of the corn- 
pass to the flag. Observing that the forward sight was nearer the 
north end of the needle than the south, I set N. in my field book ; 
and as the same sight was on the east side of the needle, I set E. in 
my field book ; and as its distance east was 43|: degrees, I set this 
number between the said letters — thus, N. 43| E.* 

Sec. 76. Compass staff. As the wind blew with considerable 
violence, I found my compass staff was too slender to keep the com- 
pass steady. I directed the axe-man to fit a stiffer one into the iron 
socket at the bottom, and the brass one at the top ; which he did 
with an interruption of but thirteen minutes of time. My staff was 
now just such as I always preferred. It consisted of a straight 
hickory sapling, two inches in diameter in the middle, with the bark 
on it, and four feet long between the sockets. From the lower end 
of the upper socket to the bottom of the sights, was six inches ; and 
the iron socket at the bottom, including its strong steel beak, was 
twelve inches. This is a suitable length for a surveyor six feet in 
height. And its great weight enables him to strike it firmly into 

* A correct surveyor always makes N. or S. the leading course, unless the course is due 
east or west; in which case the word is spelled out. He always says north or south, so 
many degrees east or west ; and plots by north and south, or meridian lines. 



53 

the ground. A tripod will never be used in the field by an experi- 
enced surveyor. 

Sec. 77. Directions to cliain-bearers. I then gave my directions 
to the chain-bearers, as follows : 

1. That neither of them should ever pass the compass, when set 
for taking the course. 

2. That the fore-bearer should carry all the stakes in his left 
hand but one, and that in his right hand, clenched around the stake 
with the chain-ring, and its point in the direction of the thumb. 

3. That he should never look back, but walk on with his eye 
upon the flag, until the hind-bearer cries down ; then he should set 
the stake firmly, and cry down, without looking back, and lead the 
chain about three inches to the left of the stake, to avoid dragging 
it from its place. 

4. That he should continue in the same manner, unless he is stop- 
ped by the flag-man, until his nine stakes are out, and until he has 
stretched his chain and finds he has no stake to set — then he cries 
tally. 

.5. The hind-bearer, having carefully arranged the fore-bearer 
by the flag, and having received all the stakes, at each of which 
he cried down ; now should drop his chain-ring, and walk to the 
fore-bearer. But he must never in any case carry forward the 
hind ring or in any way double the chain. 

6. On delivering the stakes to the fore-bearer, both must count 
them. 

7. The hind-bearer then slides one of the leather tallies from the 
left side of the knot tied in the cord behind his back, around to the 
right side of the knot, and set the fore-bearer on his way again/ 
holding the toe of his shoe precisely on the mark where the ex- 
change of stakes was made, until the hind ring comes up to it, when 
he cries down. Then all must go on as before. 

8. That if the fore-bearer comes to the flag between tallies 
(which will generally happen,) or if the distance is called for at 
any point between flags, the fore-bearer must stop when his ring 
touches the flag-staff", or other point required, and cry links. There- 
upon the hind-bearer, after carefully stretching back the chain, 
inust count off" the links back of the standing stake, and after de- 
ducting them from the whole hundred links, render to the surveyor 



54 

the number of tallies on his belt, stakes in his left hand, and odd 
links. 

Sec. 78. Offsetts. After measuring 17 chains, the hind-bearer 
gave me notice, that the pond F, was impassable. Whereupon I 
went back to the starting place and directed the ax-man to set a 
stake precisely in line with the flag at the end of the 17th chain. 
Then I set the compass at that point, and turned the sights slowly 
until the needle passed ninety degrees ; which brought them to the 
course S 46^: E. I ordered the ax-man to stand in that line at I, 
a distance sufficient to clear the pond ; which on measuring I found 
to be 8.50, where he set in a stake. There I set the compass and 
turned the sights, till the needle settled on the course of the line, N 
43| E, and ordered the ax-man to stand at i, a distance sufficient 
to clear the pond in that direction, where he set in a stake. On 
chaining to i we found the distance to be 22.50. There I set the 
compass on the reversed course of the first offset; to wit, N 46;^ 
W. Then I ordered the ax-man in that direction so far as to be 
sure to go a little beyond the original line. In this direction we 
measured 8.50, a distance equal to the offset ; which brought us 
upon the original line, where the ax-man set up a stake. These 
stakes were set up, for the purpose of correction if required ; but 
they had no influence on the survey, as the line I i was treated as 
a mere continuation of the line 1, 2, as if it had been measured 
across the pond. 

Sec. 79. Minutes. After reaching the flag at 2, I made the 
common minutes, which stood thus : 
N 43| E 47.80 

At 17. Griswold's Pond, where I made an offset at right an- 
gles, 8.50. 
At 39.50, having cleared the pond, returned to the line ; the 
parallel offset line being 22.50. 

At the corner 2, I set the compass and directed it to the flag at 
the corner 3. Finding the forward sight nearer the south end of 
the needle than the north, I set S in the field book; and as the 
same sight was on the east side of the needle, I set E in my field 
book ; and its distance cast was 421 degrees. On chaining to the 
flag at 3, the distance in measure was 18.20 ; all of which I wrote 
down as before. 



55 

On setting the compass at the corner marked 3, and finding the 
forward sight nearer the north end of the needle than the south, 
when directed towards the flag-man at corner 4, I set N in my 
field book ; and as the same sight was on the west side of the nee- 
die, I set W in my field book; and as its distance was 16 degrees, 
I wrote down N 16 W. 

Sec. 80. Heights and distances. While on this line I took the 
necessary observations for ascertaining the distance and height of 
the meeting house A, as follows : At o I called on the hind-bearer 
for the distance run on this line ; which he rendered 16.30. Here 
I directed the sight to the south-east corner of the house ; and found 
its bearing to be N 65 W. Then I directed the cross levelling 
marks on the sight to bottom of the house, and found it to be on a 
level with this station. Ranging the forward sight until the tan- 
gent marks ranged with the top of the steeple, I found, by the 
tangent scale of Kendall, before described, the angle between 
the lines of direction to the base of the house and top of the 
steeple to be 4^ degrees. At m I called for the distance as before, 
which was rendered 32.40. Here I took the bearing of the same 
corner of the meeting house and found it to be S 51 W. These 
bearings, &c., I minuted for future calculations. No other extra 
minutes were made on the line, nor on the line 4 to 5, excepting 
the places of crossing Stoney Brook. 

Sec. 81. Random line. On setting the compass at the corner 
marked 5, I could not see the corner marked 6 ; for an extensive 
piece of woodland was to be crossed. The corner boundaries were 
agreed upon by my employers and their neighbors ; but the con- 
necting line had never been run. Being obliged to run a random 
line, in order to obtain the true course, I set my employers and 
their neighbors to guess at the direction of the corner 6. Each 
stood at the corner 5 and pointed at a tree, which he supposed to 
be in the direction of the required corner. After taking the mean 
average of their opinions, the course was S 2 E. That course I 
pursued, by sending the flag-man along in that direction, as far as 
I could see him, then crying right, left, &c., until he came in the 
direction of sights ; then crying stand, he waited for me and the 
bearers to come up — again went ahead, &c., until we came against 
the corner sought. Then I turned the sights through 90 degrees, 



56 

as when taking the offset at Griswold's pond. CaUing on the hind- 
bearer he rendered 57.80 for the length of that hne. On measur- 
ing the distance to the true corner, we found the error 3.10. 

Sec. 82. Extemporaneous calculations. Not having a table of 
natural sines with me, and not being able to proceed until the course 
was corrected, I calculated the error in the course by using the 
formula 57, thus : as the distance run (57.80) is to 57, so is the error 
(3 chains and 10 links) to the error in degrees : or 57.80 : 57 : : 
3.10. The answer was found to be 3 degrees ; which added to the 
course S 2 E, gave the true course S 5 E. The true course being 
found, we returned to corner 5, and run the true line by the com- 
pass without the chain. The ax -man marked the trees which fell 
in the line, and the chain bearers were employed in throwing stones 
about the most important ones. 

Sec. 83. Ascent and descent. We run the line from 6 to 7, as 
in other cases ; and found it to be south 16.07. But in running the 
line 7 to 8 we crossed a steep hill, which required particular atten- 
tion. For the other sides being either on level ground, or mode- 
rate slopes, this side would be disproportionably long, if the line was 
run without taking any notice of the ascent or descent of the hill. 
After running a random line (which was necessary on account of 
the interposition of the hill) and finding the course to be S 63^ W, 
we returned to the corner 7, and chained the true course, on ac- 
count of taking the ascent and descent on the true line. This we 
conducted as follows. 

Sec. 84. Calculation of ascent and descent. At the foot of the 
hill, the hind-bearer rendered 3.60 as the distance run. I sent the 
flag-man to the top of the hill, where he set up his flag and held his 
hat against the mark on his staft', 5 feet from the ground — being 
equal in height to my levelling mark on the siglits, after setting my 
staff 6 inches into the ground, as usual. I took the ascent, which 
was 6^ degrees. Then running up the hill, tlie hind-bearer ren- 
dered 7.30 as the distance from corner 7. The flag-man then went 
down the hill, and set up his flag-staff and held his hat as before. I 
found the angle of descent to be 3 degrees and 30 minutes. On 
measuring to the flag, the hind-bearer rendered 14.20 as the dis- 
tance from corner 7. We then proceeded to corner 8, and the dis- 
tance rendered was 19.85. 



57 

Sec. 85. In order to reduce this line to a straight one, as if run 
through the base of the hill so as to set it down accurately in the 
field book, I made the following calculation. The angle of ascent 
and descent not being above 10 degrees, the formula 57 applied 
equally to the base and hypothenuse of a right-angled triangle. 
Therefore I said, as 57 : 3.70 (the ascending line) : : 6^ degrees 
: 0.42 (the height of the hill)— then as 6i : 0.42 : : 57 degrees : 
3.68 (the base of the hill as far as the end of the line of ascent.) 
Then I calculated the line of descent from its termination, back- 
wards in the same way — the angle being 3^ degrees — the line of 
descent 6.90, and the perpendicular the same as in the ascending 
triangle, it stood 3i : 0.42 : : 57 : 6.84, the bases added, 6.84-4- 
3.68=10.52, the length of the whole base of the hill. This de- 
ducted from the sum of the ascending and descending lifies, the 
whole line stands thus : Ascent and descent 3.704-6.90=10.60 — 
10.52=0.08. Deducting the 8 links (which is the difference be- 
tween the base line through the hill and the line over the hill*) from 
the whole surveyed line, the true line for the field book was 19.77. 

Sec. 86. Common running. We run from 8 to 9, which was 
west 17.02, and from 9 to 10, which was N 33^ E 19.00, without 
any important occurrence. Arriving at the corner 10, we proceed- 
ed, as to be related in the next section. 

Sec. 87. Oblique offset. Setting up the compass at the corner 
marked 10, and directing the sight to the flag at corner 1, (the place 
of beginning,) I found the corner to be N 79 W. But the pond G 
obstructed our direct measurement. On running to the margin of 
the pond r, which was 3.30, I found the offsets could not be made 
at right angles, as had been done at the pond F, on account of the 
pond H. Therefore I directed the flag-man to go between the two 
ponds, until he should come to a place where he judged that a line 
parallel to the course of the line would clear the pond G, which 

* I calculated this at fuli length, pari ly for the purpose of shewing the trifling difference 
iu most cases of uneven surface, and jiartly to shew the accuracy of the rule, when the angle 
of ascent and descent does not exceed 10 degree^. The jiractice of directing the bearer who 
i" at the lowest end of the chain lo raise it to tlie horizontal level, is merely professional quack- 
ery. It is impossible that any thing like accuracy should be effected in that way. Neither 
could \ rely upon this formula, if the angles exceeded 10 degrees. But I would take the an- 
gles Ln the field and calculate the error by plotting a profile of the ascending and descending 
lines, or with tables. 

8 



58 

brought him to 6. On directing the sights towards the flag, I found 
the direction to be S 59 W, and the chain bearers found the dis- 
tance 13.50. Then setting the compass at 6 with the sights on the 
course N 79 W, as before, and setting the flag-man at t, then 
measuring to the flag, we found this parallel line to be 15.70, which 
is to be added to the 3.30 measured from 10 to the pond. Now we 
return to the line at W, by measuring 13.50 on the offset course 
reversed — (that is, the offset course being S 59 W, the reversed 
course is N 59 E.) From W we continued on the course to 1. 
Adding the two measured parts of the true, and the parallel part, 6 
t, of the offset, the distance, to be set down in the field book, was 
32.60. 

Sec. 88. Survey closed. Having completed the survey of the 
outbounds of the farm, we retired to the dwelling house to make the 
calculations, which were necessary to be made before the farm 
could be divided. We dismissed all the men employed, and direct- 
ed them to return on the 2d of April. For we had occupied the 
30th and 31st of March in the survey ; and it was a full day's work 
to make the calculations. 

Sec 89. FieJd book. The following is a copy of my field-book 
as made in the field. The original entries should always be prepared 
by the surveyor, and a fair copy made for the employers. 

Field Book of an Agrometric Survey, 

For William and Robert Gardner, commenced at 7 o'clock, A. M., 
March 30th, 1801. 

Bearers, John Bemis and William Miller. Ax-man, William 
Babcock, jr. jBojO-ga^e-man, William Clarke, jr. Flag-man, Levi 
Morris.* 

Beginning boundary, stake and stones on east bank of Stoney 
Brook, N 10 W 13. from its junction with Meadow Brook, and S 
23 W 11. from the S. E. corner of Griswold's mill. 
1. N 43f E. 47.80 

* All assistants' namos should be entered in the field book; because a reference to them 
may adjust differences of opinion, long after the death of the surveyor, and their names being 
found among his field notes would save much inquiry. 



59 

17.00 offset right, 8.50, at right angles. 
39.50 offset left, 8.50, at right angles. 
"2. S 42| E. 18.20 

3. N 16 W 45.20 

16.30 meetinghouse bears N 65 W. 

Same, angle of elevation to top of steeple 4^ degrees to base 

level. 
32.40 meeting house bears S 51 W. 
37.10 Stoney Brook, 1.50 across. 

4. S 77i E. 19.00 

5.20 Stoney Brook, 1.10 across. 

5. S 5 E. 57.80 

0.00, run random line S 2 E 57.80 — woods. 
57.80, run left to the true corner 3.10 
0.00 returned and run on true line, S 5 E. 

6. South 16.07 

7. S 63^ W. 19.80 

0.00 run random line S 60 W 19.80— hill. 
19.80 run right to the true corner. 
0.00 returned to beginning and run on true course, which 

had been found by calculation, S 63^ W. 
3.60 take ascent of hill, 6| degrees. 
7.30 top of the hill — take descent of it, 3J degrees. 
14.20 bottom of the hill. Calculate here and find base line 
5 links shortest, which I deduct. 

8. West 17.02 

9. N 33J E 19.00 

10. N 79 W. 32.60 to place of baginning. 
3.50 Meadow Pond. 
Same, offset S 59 W 13.50 left. 
19.20 onset N 59 E 13.50 right. 
Compassed the farm March 31st, at 6 o'clock P. M. 
Sec. 90. Map and casting. As my employers were very desir- 
ous that the contents of their farm should be cast up with extreme 
accuracy, I cast it by the three best methods in use. 1. By sepa- 
rate triangles. 2. By reducing it to a single triangle. 3. By the 
trapezoid method, called the rectangular, or latitude and departure, 
method. 



60 

As a map is always required, the survey should be plotted be- 
fore any calculations are made. Accordingly I prepared to plot 
the survey by ascertaining how it would lie upon a piece of paper 
of a size adapted to the object. To do this, I laid down the courses 
and distances on a slate, by guessing at the course and length of 
each. In this way, I found that I must begin on the left hand side 
of my paper, about two-thirds down towards the bottom. As the 
map was required to fit a common pocket-book, I assumed for a 
scale of equal parts, 20 chains to an inch. This is too small a scale 
for accurate calculation ; therefore I drew another map with a scale 
of 4 chains to an inch. But I shall not exhibit that plot here ; as I 
proceeded with the large map in all respects as with this. 

Sec. 91. Abstract and meridians, I prepared the paper for plot- 
ting by drawing parallel meridian lines, by the opposite sides of the 
wooden inch ruler, before mentioned ; as i i, o o, u u, &c., in the 
figure. Then I copied the courses and distances from my field 
book, and reduced the distances by dividing by 20, my assumed 
scale per inch, so that I could plot by an inch diagonal scale, thus : 

1. N 43f E 47.80=2.39 

2. S 42f E 18.20=0.91 

3. N 16 W 45.20=2.26 

4. S 771 E 19.00=0.95 

5. S 5 E 57.80=2.89 

6. South 16.07=0.80 

7. S 63J W 16.80=0.99 

8. West 17.02=0.07 

9. N 33i E 19.00=0.95 
10. N 79 W 32.60=1.63 , 

Sec. 92. Plotting. Having drawn the meridian scratch-lines, I 
laid the centre of the protracter precisely at the starting point 1, 
with its straight edge on the line i i. Then I pricked the paper at 
the degrees 43|: with a sharp needle, having its eye-end set into a 
handle. Taking 2.39 from the diagonal inch scale, in the dividers, 
I set this distance in the direction of the pricked point, guided by 
the scale-rule, and drew the line 1, 2. Next I laid the centre of the 
protracter at the point 2, with its straight edge as nearly parallel to 
the meridians as I could guess ; and adjusted it 'precisely parallel 



61 

with the dividers, thus : I extended the dividers to the meridian line 
of the most convenient distance from the centre of the engraved Hne 
on the face of the bevel, and with the same extent of the dividers 
apphed one foot to the engraved hne near each end of the straight 
edge, and the other foot to the meridian line. Then I pricked the 
paper at the degree 42f , and set the distance on the line 2, 3. Thus 
I proceeded with all the lines, until the plot closed at the starting 
point. 

Sec. 93. Proof of a survey. If the plot had not closed, I should 
have measured across the farm, near the middle, to find where the 
error was probably committed. For example, if a line was run 
from corner 1, to corner 6, and should be found to agree with the 
same line drawn across the map, the error must have been commit- 
ted after passing the corner 6. But if it should have been found 
too long or too short, or if the course of the line should come out 
wrong, the error must have been committed between corners 1 and 
6 ; and the nature of the mistake could have been determined by 
the length and direction of the line. I should then have re-survey- 
ed the sides among which the error was committed. If the sides 
of the farm are numerous, several such cross lines may be run, 
sub-dividing the erroneous part of the survey, until it may be 
driven to two or three sides, before the re-survey is made. But I 
would never let a survey go out of my hands, until it had closed 
accurately, with perfectly accurate instruments. 

Sec 94. Triangular cuttings. After I had plotted the farm with 
great care, and closed the plot, I proceeded to cut it into triangles. 
I kept these objects in view while cutting it up. 1. To avoid 
making acute angles when possible. 2. To make one base serve 
for two triangles, whenever it is practicable. 3. To make the 
outside lines the bases of as many triangles as possible. When 
any of these advantages interfered, I gave the preference accord- 
ing to circumstances. The triangles A and B might have one 
base, 2, 10. But I preferred taking the trouble of two multiplica- 
tions, to save the outside line 1, 2, for a base. I saved an outside 
base in C and D. As it was hardly practicable in any other cases, 
I put E and F on the same base; also G and H. 

Sec. 95. Triangular castings. Having measured all the bases, 
and all the perpendiculars, I proceeded to calculate the areas in 



62 

the usual "Way. That is, I multiplied the base and perpendicular 
together of each triangle separately, excepting where two triangles 
had the same base in common. In such cases I added the two 
perpendiculars and multiplied both by the base, at once. After 
adding all the products and halving the sum, I multiplied it by 400, 
the square root of the divisor, by which I reduced the scale, for the 
convenience of plotting by an inch scale. This is called "raising 
the scale," by old surveyors — or "restoring true measure." Then 
I reduced the chains and links to acres, quarters, and rods, by con- 
tinually dividing by 10 (the number of chains in an acre,) and con- 
tinuing the same divisor through the reductions to quarters and rods. 
The scale may be raised, before multiplying the bases and per- 
pendiculars together, by multiplying each base, and each perpen- 
dicular, by the divisor ; as by 20 in this case. The following cast- 
ings are made upon that method. 

CASTINGS. 

Bases. Perp. Products. 

A 1 to 2, 47.80X 27.70 =1324.06 2)4750.72 

B 10 to 2, 41.80X 12.80 =535.04 10)237|5.36 

C 3 to 4, 45.20X 16.80 =759.36 4 

D 5 to 6, 57.80X 9.70 =560.66 



E ? o . r, oc ten S 5.00 > „„^ „p, 21144 

p S 3 to 7, 36.50X \ 21.50 \ =967.25 I ^^ 

G / ^ , „ „. .. ^ 13.00 
H 



I 2 to 7, 35.55X | ^'^^^ | =604.39 ^^^^^ 



A. Q. R. 

Double areas=4750.76 Ans. 237 2 5 

Sec. 96. Triangular casting reduced to a single triangle. In 
order to prove the accuracy of my calculation, I reduced the whole 
plot to a single triangle ; and then cast the contents by one multi- 
plication. Before exhibiting that operation, I will explain the prin- 
ciple with a small figure of but 5 sides, A, B, C, D, E. This method 
depends on the principle referred to in the 2d article in the 33d sec- 
tion. Having lost my brass slip on the road, I made one of a piece 
of tin with a pair of coarse shears ; and I will explain this instead 
of a better, such as I now have before me. I cut a slip of tin half 



63 

an inch wide and 9 inches long, perfectly straight. In the middle I 
drew a line lengthwise, perfectly straight and exactly parallel to its 
edges, as exhibited in the figure. The object of this slip was the 
same as the groove on the protractor ; that is, to exhibit parallels 
without defacing maps by marks, and to guide a foot of the dividers 
more accurately than could be done by a scratch on paper. 

The principle just referred to is here exhibited by the triangles 
D C B and D G B. For both stand on the same base D B, between 
the same parallel lines P, P, and 1, 1. Consequently contain equal 
areas. Therefore by extending the line A B to G, and drawing 
the line G D, there is just as much land added to the field by tak- 
ing in the area at B G. But it is not necessary to draw the parallel 
line P P. For if the dividers are extended from the angle C to the 
central line on the slip 1, 1, and then carried with the same exten- 
sion to the indefinitely extended line F G, and moved back and for- 
ward on said line, until it rests at a point (as G,) when it is found (by 
sweeping the other foot,) to be the nearest distance from the central 
line on the slip, that point will be in the parallel line P P, and at the 
angle of the new triangle. 

It will be seen, that by this operation, the angle at C is extin- 
guished ; leaving but four angles in the field. The angle E may 
be extinguished in the same way ; leaving the single triangle F G 
D. By the use of the slip, all embarrassing scratches and marks 
on the plot are avoided ; and the base and perpendicular of a single 
triangle only, are to be measured and multiplied by each other. 
Half of that product is the area of the field. 

Sec. 97. Choosing base lines. When a farm of many sides is 
to have its area calculated upon this plan, several triangles will 
grow out of each other in a manner more complicated. But if the 
slip is always laid so as to connect two corners, leaving one between, 
that one will be extinguished. One side must always be assumed 
and extended indefinitely for the base line, as F G ; every new cor- 
ner must be made on that line. The side chosen for the base line 
must not be chosen on account of its greater length ; for it will make 
the angles on it too acute for accuracy. But it must be so chosen, 
as to leave most of the plot standing upon it in its longest direction. 
After reducing the angles on one side of the plot, until they extend 
along the base line so as to make very acute angles, commence on 



64 

the other side of the field. But always work to the same base fine, 
and always begin with the angle nearest to it. 

Side A B 20.00 New triangle. 

B C 21.50 FG 49.70 

C D 26.00 ' D G 42.00 

D E 25.70 — 



E A 3^.60 ^ , 2)208.74000 

10)104.37000 
Scale 20 per inch.- 4 

1 7.48 
40 



29 920 
Acr. Q. Rd. 
10 1 29 

Sec. 98. Two base lines. It will often happen, that the angles 
formed on the base line will be too acute, even after working on 
both sides of the field. In such cases extend one of the new sides 
indefinitely, which touches the base line, and work to that as to the 
first base line. Then, when all the angles are reduced to four, 
extinguish whichever of the angles may be most conveniently ex- 
tinguished; without regard to any choice between the base line, 
whether the first or second one be finally retained. 

Sec. 99. Single triangle accurate. The advantages of this meth- 
od over that of casting the contents by separate triangles are mani- 
fest. Every step in the process is wrought by points, and one me- 
tallic line. Most errors in plotting arise while working to the 
scratch lines on paper. If the points are pricked in \/ith sharp 
round instruments, and the paper is old and of a firm texture, we 
can work to such points with more accuracy than can be expected 
from the most skilful survey. And a hne accurately engraved on 
copper, and above all on tempei'ed steel, will scarcely admit of an 
error. Considerable practice is necessary in this case, as in all 
others. 

Sec 100. Trapezoidal method. The third method which I adopted 
for proving my calculation, was the trapezoidal, or latitude and de- 



65 

parture method. It is constructed upon the following plan. Let a 
meridian line be drawn on the east or west side of the plot, so as to 
touch its ext^-eme side or corner, as the dotted line 4, 9, in the figure, 
which touches the extreme corner marked 1. Let two lines be 
drawn perpendicularly from this line, so as to touch the north and 
south extremities of the plot, as 4, 4, add 9, 9. Now calculate all 
the area included within the meridian line, the two perpendicular 
lines aforesaid, and the outside lines from the end of one of the per- 
pendicular lines to the other ; as 4, 5 — 5, 6 — 6, 7 — 7, 8 — 8, 9. 
That is, all the land, both inside of the plot and outside of it, be- 
tween it and the meridian line 9, 4. Then cast all the said outside 
part, and subtract it from the sum of both inside and outside ai'ea ; 
and the remainder will be the inside area, or true contents of the 
survey. 

Sec. 101. Accuracy of tlie trapezoidal method. The advantages 
presented in this plan are manifest on inspecting the figure. It will 
be seen that the whole may be cut into trapezoidal figures. Or 
that the north and south sides of each will be parallels, standing 
perpendicularly on the west line. All the sides but one, of each 
figure, are meridians and parallels of latitude ; consequently they 
may be calculated like latitude and departure in traverse sailing. 
Then their contents may be found by adding the departures bound- 
ing each trapezoid, multiplying them by their latitudinal distance 
from each other, and halving the product. For example, the line 
5, 5, (51.48) is the north boundary, and 6, 6, (57.15) the south 
boundary, and the line 6, 5, (57.56) the latitudinal distance, of a 
trapezoid. Then 51.48 +57.15 x 57.56-:-2=3126.3714. 

Sec 102. In and out areas. By a little attention to the figure it 
will appear, that when the measure of the latitude is from north to. 
south, it gives the length of the trapezoids both in and out of the 
plot ; and when the measure of the latitude is from south to north, it 
gives the length of the trapezoids outside of the plot. Hence it fol- 
lows, that when the southern measure is the multiplier the products 
must be added together for the inside and outside area ; and when 
the northern measure is the multiplier the products must be added 
together for the outside area, and subtracted from the other area. 
When an inner angle extends backwards, as that marked 3, the area 

9 



66 

is cast in and out twice ; but still the rule does not require any va- 
riation. 

Sec. 103. Form of arrangement. To avoid confiision, the follow- 
ing tabular formula was constructed. It will be understood by in- 
spection, after reading the preceding sections. Traverse tables, 
such as are used in navigation, are used for finding the latitude and 
departure in this kind of calculation. But it is almost as easy, and 
much more simple, to multiply the chains and links by the sine of 
the course for the latitude, and the co-sine of the course for the de- 
parture. 

Sec 104. North and south areas. In the figure the single dotted 
lines on the dotted side of the double lines, are the departures of 
northern areas. And the single lines of long dots and the long-dot 
sides of the double lines, are the departures of southern areas. 



Courses iiiid l>Hr.iiicrs I 


N. 1 S. 1 K. 1 VV ildfpl2dep| N. Area 1 S Area 


1. iN 43 ;t-4tli.^ K 47.80 


34.5,i 




3305 




33.0.i 


33.05 


1141.2165 




2. S 42 3 4tlis E 18.20 




13 37 


12 35 




45 40 


78.45 




1048.8765 


3. N 16 W 45.20 


43.45 






12.45 


32.95 


7-35 


3404.3075 




4. S 77 1 4th E 19 00 




4.19 


18 53 




51 48 


84-^3 




3.53.7617 


5. S 5 R 57,80 




57.56 


5.67 




57.15 


ias.G3 




6252 7428 


6. South 16.07 


i iC.07 






57.15 


114 30 




1836 8010 


7. S 63 1-2 W 19 SO 




8.84 




18.71 


38.44 


95 59 




845.0156 


8 West 17.02 








17.02 


21 42 


59 86 






9. N 33 1-2 E 19 00 


15 84 




10.49 




31.91 


53 33 


844 7472 




10. .\ 79 W 32 60 


0.21 






3191 


00 00 


31.91 


198 1611 






100.031 100.03 


80.09 


60.09 




5588.4323 


10337.1976 






5588.4323 




4748.7053 















r237A 


cr. 1 Qr. 


30 Kds. 



Sec. 105. Merits of the three methods. Having cast the con- 
tents of the farm by three methods, all of which I have long used 
in practice, it may be proper to express my opinion on their relative 
merits. I say, most decidedly, that the method of plotting and re- 
ducing the plot to a single triangle, is the best known method, for 
ordinary cases of farm surveying. For smooth even plains, and 
for city lots, the trapezoidal method is best. I have no room here 
for my reasons at length. But who will not perceive at a glance, 
that uneven land requires an averaging method, which is not prac- 
ticable by any method but by accurate plotting 1 Moore himself 
(the inventor of this method, of whom I learned it personally before 
he published it,) acknowledged that defect in his latitude and depar- 
ture method, as he named it. 



67 

Sec. 106. Heights and distances. Having completed the survey 
of the farm and cast its contents, it remained to calculate the dis- 
tance and height of the meeting house, observed while running the 
line 3, to 4, before I proceeded to divide it. In the field book un- 
der section 89, it appeared that observations were taken at two 
stations, o and u, which were 16,10 apart. At one station the bear- 
ing being N 65 W, and at the other S 51 W, it appeared by inspec- 
tion that the angle at o was 49 degrees, and at m 71. These sub- 
tracted from 180 left the angle at the meeting house 60 degrees. 
By section 44, the length of the line from the house to o was found 
thus: as .86603 (the sine of 60) : 16.10 :: .94552 (the sine of 
71) : 17.57. Then I found the height of the meeting house thus: 
the angle of elevation being 4f , I subtracted it from 90 degrees, 
and it left 85| degrees for the angle at the top of the steeple. Then 
as .99692 (the sine of 85 J degrees) : 17.57 (the distance from the 
meeting house to o) :: .07846 (the sine of 4J degrees) : 1.38=91 
feet. That is, the distance was 17.57, and the height 91 feet. 

Sec. 107. Heights and distances geometricaUy. I made a calcu- 
lation also, by a plot thus : I laid down the courses and distances, 
from the points o and u, indefinitely, and measured from o to the 
point of the intersection of the lines ; which gave the distance. 
Then laying down that distance as the base, raising an indefinite 
perpendicular on one end, and laying down an indefinite line from 
the other end at an angle with it of 4J degrees, the point of inter- 
section gave the height. But I laid down these plots by a much 
larger scale than I had used for plotting the farm. 

Remark. If I had made frequent trials along the line 3, 4, so 
as to have found the point where the beaiing of the meeting house 
would form a right angle with the line, both calculations might 
have been made by the formula 57, with and without reducing. 

Sec. 108. Division of land. Before making calculations for 
dividing the farm, I inquired whether any points were fixed upon. 
I was thereupon directed so to divide the farm that William should 
have the north end, and 10 acres less than Robert ; and that one 
end of the division line should be on the line 1, 2, at z, the margin 
of the pond, where we made the first offset. On looking over the 
map, [see section 81] I found that the corner 6, would be nearer to 
the termination of a division line, starting at z, than any other cor- 



68 

ner. I drew a line through z, 6, indefinitely, for a base upon each 
side of which a triangle was to be formed, as described in section 
ninety-six. By this operation I found 1287.50 on the north side, 
and 1086.88 on the south side, of said base line. Consequently 
the east end of the division line required to be moved 6.60; far 
enough to include half the difference, 100.31 added to half of the 
10 acres=150.31. 

Sec. 109. Dwision hy a triangle. Here I was obliged to intro- 
duce a new application of the section 52, one, two, relating to the 
areas of parallelograms and triangles. For as a triangle is half a 
parallelogram, it is manifest, that if the quantity of land to be taken 
from the north side (5 acres and half the difference between sides) 
was doubled and that sum divided by the line z, 6, (found to be 
45.50) the quotient would be the perpendicular of the triangle to 
be added; thus, 150.31x2-^45.50=6.60, the distance to which 
the east end of the line z, 6, was to be removed north. I drew the 
line z, a, and measured its course and distance on the map, which I 
found to be S 87^ E 44.00. Then I went to the field, run and 
marked said line and set up the necessary boundaries. I gave 
each an entire field book of his separate share, after altering the 
line 6, 5, to accommodate the divisions to the divided field notes. 

Remark 1. It must not be forgotten, nor overlooked, that after 
finding, that 300.62 was the difference between what was made by 
the assumed division line, z, 6, and what was required, but half 
that sum was to be taken from the north part — as one acre taken 
from the one part and added to the other, will make a difference of 
two acres between the parts. 

Remark 2. This method of dividing a farm, and casting the 
parts separately is an excellent method for proving a survey. A 
farm of numerous sides may be thus divided into three or four 
parts for more perfect proof of accuracy. It may be farther ob- 
served, that if we were sure that the survey and calculations were 
correct, a farm might bs divided by a ceJculation made on one side 
of the cross line only. 

Sec. 110. Road surveying. Road surveying, when nothing 
more is required than courses, distances, and notices of objects, be- 
longs to Agrometry. But when ascents, descents, dug-ways to be 
made, beds to be raised, &c., are to be calculated, it belongs to 



69 

Engineering, described hereafter. In road surveying, however, the 
field notes are kept differently. The chain is carried directly on, 
without starting anew at the angles or turns in the road, as in farm 
surveying. But the change of course is always set down, like 
other incidents — and the leathern tallies are carried as directed in 
section 77, until the seven are moved, and then another tally is run ; 
when the hind-bearer will be reminded that eight tallies are out, by 
finding no tally to slide. Then he notifies the surveyor that one 
mile is completed. This is entered in the field book, unless the 
hind-bearer is also entrusted with the mile entries. 

Sec. 111. Road, field look. The following is the form of the 
field book of a common road survey. 

Field look of a Road Survey from Catskill Village to Catskill 
Mountains. 

Beginning at the west end of Benton's bridge. 

0.00 S 85 W. 

10.21 Meiggs' house, left. 

31.20 Gordon's house, I'ight. 

36.07 Cross road from Catskill to Fall mills. 

42.90 S 72 W. 

140.31 Long Swamp. 

247.60 N 64 W. 

317.90 S 86 W. 

560.41 Kisketam flats, at Coat's land. 

640.27 Greene Patent ledge in M. Lawrence's range. 

Same N 79 W. 

720.00 Main Catskill Mountain. 

in. OROMETRY. (Line surveying, reviving lost lines.) 

Sec. 112. Running lines from a plot. \n Orometry^ihe courses 
and distances are always given ; but the objects of the survey are 
various. Sometimes the out-bounds of large tracts of land are run 
and established, and the tracts are cut into lots on paper ; then the 
business of the surveyor is to run, and mark, the boundaries. In 
such cases the surveyor must be very careful to plot from level 
out-boundaries ; and, in cutting up the tract into lots, he must be 



70 

eqimlly careful. For without such care, future tenants, or pur- 
chasers, will have good reason to complain of uncertain boundaries. 
Sec. 113. Reviving old lines. But after a district of country is 
inhabited, the most common cases in Orometry are, the running and 
marking of old lines, lost by negligence. In such cases, if any one 
corner is remembered and can be precisely located, all the other 
corners and lines can be found. Much experience, however, is re- 
quired for searching out old lines. If several boundaries accord 
with each other, this accordance has great weight with honest far- 
mers, also with jurors, in fixing the lines. For no one will doubt 
the correctness of lines, when several sides of a farm coincide with 
the written courses and distances, even if no well established corner 
can be found. 

History of an Orometric Survey. 

Sec. 114. Reviving lost corners. On the 21st of June, 1803, I 
commenced the survey of the tract of land, called Disc's patent, in 
Schoharie county. New- York. It had been surveyed in the year 
1743,* sixty years before I surveyed it. The boundaries were lost 
in most cases ; and the proprietor of Scott's Patent (which belonged 
to John Livingston, Esq., whose agent 1 then was) accused Mr. 
Rechraeyer, the proprietor of Disc's patent, with encroaching upon 
his tract. A survey, therefore, became necessary. But there was 
not a corner boundary established. Several side lines were pretty 
well marked. 

Sec. 115. Settling magnetic variation. The first point to be set- 
tled was, the variation of the needle from 1743 to 1803, a period of 
60 years. Having been told by the Surveyor General, De Witt, by 
Messrs. Cockburn, Wigrai^i, Trumpbour, Van Alen, and Savage? 
of the State of New York; also by Samuel Moore, of Salisbury, 
Connecticut, that the north point of the needle had been appx'oaching 
the meridian for a century, at the rate of nearly one degree in twenty 
years, I adopted that fact for my rule of calculation. 

Sec. 116. Proving tlie variation. In order to make the allow- 
ance, without the possibility of mistake, I concluded to run thelong- 

* I surveyed this tract at Uiat time, (June 21st, 1803,) but it is too large to present the whole 
survey here. Theretore I leave out several sides, and I leave out some precise dates of the 
patent also. 



71 

est boundary line first, on which any marked trees could be found. 
Not being able to find a corner, I set down the compass at a place 
on the line where I found several marked trees nearly in a range 
apparently agreeing in direction with the given west line ; which 
was S 2|- E. I let the needle settle on the given course and then 
turned the north point east thi-ee degrees with my finger; as it had 
moved east three degrees in sixty years; according to the opinions 
of our most experienced observers. This brought the north point 
half a degree east of north ; consequently the course to be run was 
S ^ W. Then I turned the sights (after replacing the glass cover) 
so that the south point of the needle rested ^ west. I run on that 
course by directing the flag-man as usual, until I became satisfied 
that this was probably the true line, from the number of marked 
trees which coincided with it. 

Sec. 117. Proving old marks. Lest thesa should be spurious 
marks, I directed the ax-man to cut in above and below several of 
them until dead wood appeared ; and then to split out blocks at the 
depth of the dead wood. Here we found, more or less distinct, 
J. D. (for John Dise) 1743. On the outside we found a distorted 
D on several trees. On counting the grains (concentric cylinders) 
we found from 40 to 50 very distinct, in several cases ; and indis- 
tinct ones, which might supply the remainder of the 60 required to 
answer to the 60 years. As all the new layers of wood are intro- 
duced between the bark and wood in the form of mucilage (cam- 
bium) which hardens into thin concentric layers, all the grains ap- 
pear as if no mark had ever been made on the tree ; excepting 
those near the original interior mark, and the distorted outside mark. 

Sec. 118. Finding old corners. As no corner could be found 
at any place which could be relied on, and as this line and the 
north line run through ancient forests where numerous marked 
trees remained, I concluded to run these lines until they should 
cross each other; and then to assume the crossing for a starting 
corner. By tracing all the hues from that corner, if I found a 
coincidence with similarly marked trees on several sides, I should 
believe I had run the old lines truly. On making the trial, I suc- 
ceeded to the satisfaction of all parties concerned. But the allow- 
ance of one degree for 20 years was certainly too much by several 
minutes. 



72 

Remark. At the present time (1837) it is well known that the 
needle was stationary about the beginning of this century in this 
district ; and that it is now on the retrograde. 

Sec. 119. Aberrations of needle often imaginary. In Catskill' 
Greene county, New York, I run the boundary lines between two 
tracts called Row Patent and Greene Patent, in the spring of 1808. 
I could not close the Row Patent by four chains ; though I plotted 
it several times with particular care. I concluded to resort to a 
survey, with a view to ascertain the side on which the error was 
committed. By comparing the descriptions, finding corners with 
cross lines, as in the Dise's Patent, I was able to settle with satisfac- 
tory proof, that the fore-bearer must have lost 4 stakes, which the 
hind-bearar omitted to correct. As the survey had been made 70 
years before, none of the assistants could be found ; though all their 
names were found in the field book of the surveyor, George Met- 
calf, Esq. While running one of these Hnes my needle varied ; and 
I was obliged to send home for another compass which traversed 
well. It was at this time, I first observed, that my needle varied in 
that compass always when the sights were directed nearly as on 
that course. This led to the discovery, that most, if not all, the de- 
viations of needles, which are ascribed to the attraction of iron 
mines, are caused by fine grains of iron, left in the card or limbs of 
the compass.* 

IV. UDROMETRY. {Marsh and aquatic surveying.) 

History of an Udrometric survey. 

Sec 120. River and harbor survey. I was employed to survey 
a trunk of Fludson river between Albany and Rensselaer counties, 
for the purpose of ascertaining which side of a middle ground ought 
to be selected for the purpose of improving the best channel. On 
the east side the channel was known to be the deepest ; but the bot- 
tom was rocky and difficult to excavate. But the west side was 
loose gravel, easily excavated. 

* See Sillinian's Journal, vol. 12, p. 14, where I published several facts relating to that 
subject ; and proposed that the needles should be very sharp and capped with brass or silver. 
By this means the steel point of a needle may be kept at a little distance from the limb, and 
defended from rust at the sharp point. It is found to be perfect in practice. 



73 

Sec. 121. StaJdng out the ground. First I caused my assistants 
to set stakes at all the turns on both shores of the river, and at all 
the turns on the middle ground. Then I took soundings along both 
channels. Wherever I measured the depth, I set up a stake by 
tying a stone to one end by a very short rope, for an anchor. On 
the other end I marked the depth, and tied colored rags to it, that I 
might readily see it from shore. The spots on the river-plot repre- 
sent the stakes, and the figures their various depths in feet. 

Sec. 122. Base lines. I run two base lines on shore, where I 
could find the best ground ; to wit p, s, and iv, t. On these lines I 
made marks, at p, q, r, s, t, ?/, v, and w. I chose these places so 
that I could see all the stakes, each at two stations ; at both of 
which a compass was set. 

Sec. 123. Taking hearings. Setting the compass at _p, I directed 
the sights to 21, 5, z, 18, e, and 12 ; and I noted down each bearing, 
numbering them from north to south. Removing to q, I took the 
bearing of all the stakes which were nearest to this station. In this 
manner I continued until I had taken the bearings of all the stations 
on both sides of the river — each from two stations, by one or two 
compasses. 

Sec. 124. Points of intersection. It is manifest, that (as the course 
of the base lines had been taken, and the distances from station to 
station measured,) the distance of the stakes from each other, and 
from the base lines, will be indicated by the intersection of lines 
drawn according to their bearings. And as numerous bearings 
were taken at each station, the plotting is easily performed ; for all 
the bearings from one station may be marked off without taking up 
the protractor. 

Sec. 125. Lines of the shores. By measuring at right angles 
from the base lines to the staked turns on the shore, or to any other 
accessible object, they may be laid down by erecting perpendiculars 
of the measured lengths, at the places noted on the' base lines. 
These places may be connected so as to give the true form of the 
shore, &c. 

Sec. 126. Notations on the map. The position of the stakes, 
with the depth of the water at each, was marked on the plot or map, 
at the intersections, as shown on the map. Notes relating to the 

10 



74 

bottom, were made in the field notes, with references to the stake 
as marked down. 

Remark. AH harbor surveys may be made upon this general 
plan. Also any surveys of ponds, marshes, &c. But particular 
cases require a judicious plan, adapted to its pecuUar circumstances; 
and no set of rules can apply in all cases. The surveyor's judg- 
ment will always give character to such surveys. 

STATICS AND DYNAMICS ; 
As far as they are necessary to Civil Engineering. 

Sec. 127. Statics.* The science of gravitation or pressure, 
while bodies are restrained from motion ; as the mechanical powers 
when in equilibrio, or the compression of metals or timber, under 
heavy weights. The construction of bridges, piers, roofs, dams, 
flumes, &c., requires to be designed according to the laws of 
statics. 

Sec 128. Dynamics.! The science of motion, or moving force, 
applied or applicable to bodies when free to receive motion. As 
the power applied to a lever while raising a weight — the power of 
gravitation in giving motion to a carriage down an inclined plane — 
the power applied by means of a pulley in raising casks, &c. 

When Statics and Dynamics are applied to water, they are deno- 
minated Hydrostatics'^ and Hydrodynamics.^ 

Sec 129. Hydrostatics, applies to water, while its pressure is 
resisted ; as the strength of planks and their fastenings on the frame- 
work of flumes, must be in proportion to the square root of the head 
of water. 

Sec 130. Hydrodynamics, applies to water in motion ; as the 
velocity of water in a raceway will be in proportion to the inchna- 
tion of the plane of its descent. The various instruments and struc- 
tures employed in hydrodynamics, are called Hydraulics. 

* Statos, Greek, (from istemi,) standing, being stationary. 

t Dunamis, Greek, power, efficiency. 

t Udor, Greek, water, prefixed to States. 

§ Udor, Greek, water, prefixed to Dunamis. ^ 



75 



FALLING BODIES. 

Sec. 131. All bodies, both solid and fluid, are accelerated equally 
by the attraction of gravitation. This is demonstrated by what is 
called the Droppers experiment with the air pump. The droppers 
receiver is a glass cylmder, usually about 18 inches in height. The 
air being exhausted, a feather and a piece of lead are dropped from 
top to bottom, by means of an appropriate apparatus. As the fea- 
ther reaches the bottom as soon as the lead, it follows, that it is the 
atmospheric air only, that causes the difference of velocity in all 
common cases of falling bodies. 

Sec 132. After numerous and extremely exact trials, it is found 
that all bodies fall 16.2 feet in one second, in a vacuum. Also, of 
cou S8, that in falling one foot, a body acquires such an increased 
velocity, that should its increase be suspended at the end of the foot, 
it would thence forward move at the rate of 8.1 feet per second. 
But the continued increase carries it 16.2 feet in one second from 
its starting. This principle will be more clearly illustrated with 
water-pressure. 

WATER, 

As AN Agent in Engineeeing. 

Sec. 133. Students should learn, from trial, to estimate water 
with facility, by weight, cubic measure, and common liquid mea- 
sures. A pint of pure water weighs one pound.* The weights 
and measures adopted in this treatise, will be 2000 Jfes. neat weight 
for a ton, in accordance with the revised laws of New York. Also 
a cubic foot of water to 60 ifes., and 28.8 cubic inches to a pound or 
pint of water. Let every student measure and weigh water by 
using common pails, cups, tubs, &c. Let vessels of all forms be 
measured by the inch as an exercise ; and the correctness of the 
measures proved by weight and a sealed liquid measure. 

* It is uncertain which was first adopted, the pint measure or pound weight ; but it is evi- 
dent that accident did not cause their agreement. 



76 

Sec. 134. Hijdrometers should be immersed in water also ; par- 
ticularly Baume's Areometer. 

Manner of using Baume's Glass Areometer* in ascertaining the 
specific gravity of liquids. 

In constructing this instrument two stationary points are assumed ; . 
and if you have none at hand, these points may be found as follows. 
Take a slender glass tube, with a hollow bulb at the bottom. Put 
into the bulb mercury or fine shot, until you sink it in pure water 
almost to the top. Mark the zero point at the surface of the water. 
Then weigh 85 parts of water and 15 parts of table salt (muriate of 
soda.) After the salt is perfectly dissolved in the water, bring the 
temperature to 57° of Fah. Immerse the tube in this solution, and 
mark the point at the surface of the water, for the lower termina- 
tion of 15 degrees. Being equally divided into 15 parts, these parts 
may be assumed as standard measures for any series of tubes (one 
ending where another begins,) for taking the relative specific gra- 
vities of liquids from the heaviest sulphuric acid to the lightest ether. 

Water used in taking specific gravity of Solids. 

Sec. 135. To be familiar with taking the specific gravities of ma- 
terials for construction, is often of great use to persons in all other 
situations in life, as well as to engineers. 

Illustration. Tie a strong silk thread or silk twine around a piece 
of marble weighing seven or eight pounds. Weigh if carefully, using 
balancing quarter-ounce or half quarter-ounce weights ; so as to 
bring it to an even ounce weight on the steelyard bar. Then weigh 
it in water, sinking it so as to be wholly about half an inch below 
the surface of the water. Next subtract its weight in water from 
its weight in air — take this remainder for a divisor, and its weight 
in air for a dividend ; and the quotient will be its true specific gra- 
vity. As the weight in air is 8 ife. 7^ oz, ; weight in water 5 ife. ^ oz. 

* Mraios, Greek, slender or delicate ; and metron, a measure. 



77 

In air, 8 ife. 7^ oz. = ft. 8.453 
In water, 5 ife. i oz. = ife. 5.016 



Rem. 3 ife. 7 oz. = ife. 3.437 

.3.437)8.453(2.459 spe. grav. ; that is twice and 
6.874 about ^Yo heavier than an 

equal bulk of water. 

15790 
13748 



20420 
17185 

32350 
30933 

In this manner the solidity of materials for construction may be 
readily obtained — and it is preferable to the usual practice with 
grain- weights, for coarse materials. 

Hydrostatics. 

Sec. 136. Make a cylindrical bellows, by cutting two circles of 
thick board 10 inches in diameter, and nailing to the outside rim of 
each, with broad headed tacks, a hollow cylinder of leather. When 
finished it will present a leathern cylinder of strong calfskin, 10 
inches in diameter and 8 inches long. Set in the middle of the top 
board a leaden tube of about the fourth of an inch in calibre, and 3 
or 4 feet high. Let the top fit into a glass tube, 5 to 10 inches long, 
by a bandage of tow. When used, the leather needs to have been 
soaked in water several hours. Fill the cylinder with water through 
a plug-hole in the top board. Lay a weight on the top board, or let 
a student of suitable size stand on it, so that the water may rise into 
the glass tube. On measuring the height to which the water rises 
in the glass tube from the top board, and making the proper calcu- 
lation, this result will be found : the weight set on will precisely 
equal the weight of a cylinder of water, 10 inches in diameter, of 
the height of the water in the tube. Hence it follows, that water 
presses according to its height ; not according to its quantity by 
measure or weight. Therefore were it not for the impossibility of 



78 

maintaining the perpetual supply of water, a tube of an inch calibre 
would be sufficient for moving the machinery of an extensive factory, 
under a hundred feet head, supplied from a small reservoir or tub. 

Hydrodynamics. 

Sec.. 137. After this is demonstrated, that water pressure depends 
on its height, and the weight of a standard measure of water is 
ascertained, we must determine by trial, what measured velocity 
will be given to water by a measured head. It was before stated, 
that trial has shown that a measured pint of pure water weighs a 
pound. 

Sec 138. Trial has also shown, that under one foot pressure,- 
water will be forced through a lateral aperture with a velocity of 8 
feet and a tenth, per second, in a vacuum — probably it will be cor- 
rect in practice to say 8 feet per second in the atmosphere, at the 
. precise point of effusion. This trial prepares us for the universal 
~ rule which governs in all cases of motion by gravitation. 

Sec. 139. The increased velocity of water effused, and of falling 
solids, is as the square root of the head of water, and as the square 
root of the distance through which solids fall. Taking 8 feet of late- 
ral effi;ision per second for the first foot, and the increase from that 
zero (if it may be so called,) as the increase of the square root of 
the head, and the increase of the distance fallen through in the case 
of solids, we arrive at results of vast importance in engineering. 

Sec 140. Illustration. A spacious flume has 25 feet of water 
in depth, with five apertures or gate-holes, of one square foot each. 
The centre of the first gate-hole is one foot below the top of the 
water — the second 4 feet — the third 9 feet — the fourth 16 feet — the 
fifth 25 feet. The square root of one is one, and the lateral effusion 
of water is 8 feet per second, as demonstrated by trial. The square 
root of 4 is 2, and the lateral effusion of water is 8x2=16. The 
square root of 9 is 3, and the lateral effusion of water is 8x3=:24. 
The square root of 16 is 4, and the lateral effusion of water 8 x 4= 
32. The square root of 25 is 5, and the lateral efflision of water is 
8x5=40. All intermediate heights of head may be calculated in 
the same manner. That is, extract the root of the height given, in 
feet and decimals of feet. Multiply that root by 8, the velocity of 
the first second of pressure. See the annexed diagram. 



79 



^ 



■^ 
s 
















o 










CO 




















T-i 










00 










o 

o 


l-H 


II 

00 

X 


II 

OD 

X 

CO 


II 

00 

X 


II 

00 

X 


Pi 













w 



>. 



fo 



be 



5b 



^ 






J3 



>-> 



> 


^ 




<u 


^ 


s 


o 


rr! 


.s 


p 






CP 




Pi 


<15 


■+-J 


^ 


<v 


-♦-J 


^ 


O 






o 

l-H 


g 






o 


^ 


a 


^< 




frt 


in 


01 




a 



80 

Remark. Though the distinction between Statics and Dynamics 
is truly philosophical, it is inconvenient in its application to the Ma- 
thematical Arts — more especially so in a concise plain treatise, 
wholly devoted to practice. Olmsted's Compendium of Natural 
Philosophy is particularly recommended to students in Engineering, 
who have time to discipline their minds for a more systematic view 
of Mechanical Science. 

Water unber the influence of Atmospheric pressure. 

Sec. 141. The atmosphere presses upon all bodies on the earth, 
at tide-water level, at an average of about 15 Ifes. on every square 
foot. Water is held down with a force, very unexpected by the 
student, until he makes the common trial, as follows : Boil water 
in the open air ; and introduce the bulb of a thermometer at the 
moment of boiling. The mercury will rise to about 212° of Fah. 
Take off the atmospheric pressure perfectly, and it will boil at 67° 
As such accurate apparatus as this experiment requires is not com- 
mon, an approximation to it must be used, which will satisfy every 
student. Put a gill of water in an oil-flask (I mean those Florence 
flasks, with a flag covering woven on them.) Cork it perfectly 
tight ; and let one inch of the top of the neck be well wound with 
waxed thread, to prevent splitting by suddenly forcing in the cork. 
Before the cork is put in, boil the water for about one minute. 
This will force out the air, mostly. Thrust in a compact, soft, vel- 
vet cork (as the best are called,) while the water is boiling. Now 
let tlie water cool down to blood-heat ; which may be known by 
applying the hand of a healthy person. If he can scarcely per- 
ceive any warmth to the hand on applying it to the bottom of the 
flask, the temperature is about 98 degrees, that is, 66 degrees above 
freezing. This is 114 degrees below boiling. As it is but 180 de- 
grees from freezing to boiling, and as 114 is 66 above freezing; 
even this imperfect experiment proves, that almost half of the heat 
required for boiling water, is applied in resisting atmospheric pres- 
sure. And accurate experiments prove that more than two thirds 
of the heat is employed in counteracting the pressure of the atmos- 
phere. 

Sec 142. Atmospheric pressure is most perfectly exhibited, by 
a tin tube about 34 feet in length. Let this be closed at one end> 



81 

and terminated by a few inches of glass tube. Fill it with water* 
and invert it in a vessel of water. The weight of the atmosphere 
will press upon the surface of the water in the vessel every where 
alike ; but finding no resistance in the tube, (the air having been dis- 
placed by filling with water before inverting it,) the water is raised 
between 30 and 34 feet, until it counterbalances the weight of the 
atmosphere. A cheaper and more convenient method is, to use a 
torricellian tube, or (which is the same thing) a barometer. The 
height to which mercury, is raised in the tube by atmospheric pres- 
sure, may be readily applied to water by calculation ; reckoning 
mercury as 13J times as heavy as water. If the mercury rises 
29.92 inches, multiply this by 13.5; it gives 403.92 inches — divided 
by 12, it gives 33.74 feet. This calculation is important in fixing the 
upper valve of a pump, in regulating water works, 6z;c. 

Sec. 143. Illustration. A pump maker had practised setting up 
pumps near the tide-water level on Hudson river. He was em- 
ployed by a potash manufacturer in Castleton, Vermont, to erect a 
pump. He adopted his accustomed rule for fixing the upper valve. 
On a warm damp misty day (soon after the pump was set up) it was 
found that water could not be raised. The lecturer to the Medical 
class on Natural Philosophy, was called on by the proprietor (who 
was a trustee of the institution,) for an explanation of this strange 
phenomenon. It was found, that when the air was exceedingly 
Ught, by warmth and by being surcharged with vapor, mercury 
would not rise in the torricellian tube to a sufficient height to carry, 
by calculation, the water to a sufficient distance above the upper 
valve to give play to the piston. Lengthening the piston-rod and 
lowering the upper valve, immediately corrected the embarrass- 
inent. 

Sec. 144. Illustration. I have been told that the head of the wa- 
ter works at Hudson, N. York, (two miles out of town,) was more 
than 30 feet lower than a high ground, over which the pipes were 
laid. I have often seen the head of the works in the present cen- 
tury, which were in good order ; but this is said of their commence- 
ment some 40 or 50 years ago. True or false, the principle may 
be illustrated by supposing it true. It is said, that in damp warm 
weather, the water would not run. This is in accordance with the 
laws of pressure. Atmospheric pressure will carry Avater, in close 

11 



82 

air-tight pipes, over hills from 30 to 34 feet above the spring head, 
near tide-water level, as may be demonstrated to students, by the 
barometer or torricellian tube before described. 

GONATOUS FORCES, 

APPLIED TO BRIDGES AND OTHER FRABIE WORK, AND TO STONE AND 
BRICK ARCHES. 

Sec. 145. Gonatous forces (from gome, gonatos, Greek, angular 
flexions, like the knee joint,) are applied by means of angular flexions 
of ropes or bars ; as the genicular braces of carriage calash-tops, or 
the sailor's funicular advantage in hoisting a sail by springing the 
halliards out from the mast. The lavi^s of pressure applied to bridges, 
arches, &c., are better explained on the gonatous principle than by 
any other means, as follows : The weight of a bridge may be made 
to res!: on a point; and that point to rest on a single pillar of stone 
or wood. The pressure would then be downwards, and single in 
direction. A ton (2000 ife.) would press directly on the earth at 
the foot of the post. But if it pressed on the meeting point of a pair 
of rafters, here would be a resolution of force into two equal- parts ; 
one half in the direction of each rafter. And the pressure being in 
the line of the woody fibre of the rafters, it would not crush them 
without very great weight. This pressure would not be directed 
downwards, as if on a pillar; but would pi-ess outwards as well as 
downwards, and tend to spread or separate whatever supported the 
rafters at the foot. If the angle at the meeting of the rafters should 
be 126° 52', the spreading pressure would be double the downward 
pressure, according to the law of diagonal lines. For each rafter 
would be the diagonal of a parallelogram, twice as long horizon- 
tally, as deep vertically. This is the common method of estimat- 
ing roofs, bridges, &c., also arches, by estimating short chord lines, 
separately taken. But the same results may be produced by the 
application of the gonatus principle, called the genicular and funicu- 
lar power ; and it is more simple and of extensive generality. It 
will be understood by a reference to the knee braces of chaise tops, 
or better by inspecting the printing press, to which this power is 
very advantageously applied. 



83 

Sec. 146. As the genicular* miAfunic\dar'\ powers are very im- 
portant in their practical application, let students make the foUovvuig 
trials. Fasten two pullies against a wall about 8 feet apart. (If 
you have no pullies, smooth wooden pins, half an inch in diameter 
and five or six inches long, may be substituted.) Draw a very 
flexible cord or rope over them, and attach weights to their pen- 
dant ends. Let the pendant ends hang down from each pin about 
4 feet. Now apply different weights to the middle, and to the ends 
of the rope. . By thus flexing the rope in the middle, this principle 
will be demonstrated. As half the weight applied to the middle of 
the rope is to one of the end weights, so will be the descent of the 
middle of the rope, to the length of the section of it between the mid- 
dle weight and pulley. It will be perceived, that this is the law 
of the inclined plane, inverted. 

Sec. 147. Students can easily be made to realize the almost uni- 
versal application of the law to be deduced from this experiment. 
For though the flexion of the rope is downward, the principle is the 
same. At the commencement of the flexion of the rope, one pound 
will raise hundreds. But let the flexion continue to be increased, 
until the angle is reduced to 90°, and the farther flexion will require 
great weight. So if the rope was changed to a jointed flexed bar 
of iron, and that tui'ned with the angle upward, it would resist great 
weight, if placed upon the outer point of the angle. The law of the 
inclined plane would perpetually apply. That is, the weight would 
press in a perpendicular direction proportioned to its pressure in a 
lateral direction, as the distance of the angle from a straight line 
with the ends of the bar, to the length of a side of it. 

Sec. 148. The application of this law to frame- work or to stone 
arches, may be farther illustrated by setting the pullies or pins in 
the wall or ceiling, so that one shall be 4, 5, or 6 feet higher than 
the other. Then fasten the middle weight to the middle of the rope 
by a piece of twine. Here all the weights will hang down, present- 
ing a fair exhibition of the action of the law of gravitation. And 
the same law will apply to the flexion from a straight line compared 
with the length of the line between the middle and the pulley. The 
student should be taught to plot arches in short chord lines ; and 

* Genicular, from genu, Latin, a knee, 
t Funicular, from funis, a rope. 



84 

then to lay down each according to the laws of the funicular power ; 
transferred to the genicular, as above described. 

Sec. 149. An exhibition of the application of the genicular power 
to arches, &c., may be cheaply made, as follows : Get out about 12 
strips of wood, like common rulers, about an inch and a quarter in 
width, one third of an inch in thickness, and a foot in length. Let 
these be united in parallel pairs, and the six pairs be joined by a two 
inch piece, forming a free double joint with screw rivets. This six 
feet of six free joints, may be bent into every form of arch ; and, by 
tightening the joints with the screws, the arches may be made to 
exhibit all the various effects of pressure. By hanging common 
cast-iron weights on various parts of the various arches into which 
the joints may be bent, every view may be reduced to mathematical 
calculations. 

MECHANICAL POWERS. 

Sec. 150. The mechanical powers are, elementarily, but two — 
the Lever and the Inclined Plane. 

Lever is subdivided into Lever proper, Wheel and Axle, and 
Fully. Inclined Plane is subdivided into Inclined Plane proper, 
Wedge, and Screw. 

Sec. 151. Lever is prying, when the fulcrum is between the 
power and weight. 

Lever is lifing, when the weight is between the power and ful- 
crum. 

Lever is radial, when the power is between the weight and ful- 
crum. 

Wheel and Axle is a perpetual lever, either radial or prying. 

PuLLY is a perpetual lifting lever. 

In all levers the power is to the weight inversely as the side to 
which 'it is applied is to the side applied to the weight — the bar, or its 
equivalent, being balanced. 

Sec. 152. Inclined Plane Proper. In all inclined planes the 
power is to the weight as the height of the elevated end is to the 
length of the plane. 

Wedge. The power is to the resistance as the thickness of the 
head, to the sum ^of the length of the two slant sides. But the use- 
fulness of the wedge does not depend much on its advantages as a 



85 

power, when applied to splitting rails, &c. But it gives direction 
to a very great degree of momentum, acquired by swinging a beetle, 
sledge, &c., and suddenly expending all of it on the head of the 
wedge. 

Screw. The power is to the resistance as the distance between 
the threads of the screw is to the length of a circular thread. The 
screw is chiefly useful in giving a very advantageous direction to 
the lever, as in case of the cider-press. It is also useful in making 
delicate mathematical instruments ; as an index on its head, or 
handle, may indicate the smallest possible degree of movement. 

Remark, These general principles are sufficient ; for students 
must see the instruments and use them. It is impossible rightly to 
understand them from mere description. Olmsted's Compendium of 
Natural Philosophy may be profitably consulted on the subject of 
these powers. 

ARCHITECTURE. 

Sec. 153. Though Architecture is made up of artificial mate- 
rials, it is truly a science. The historical origin of some of the ele- 
mentary principles is rather obscure ; but in most cases their history 
is known. The early ornamental buildings were of stone and brick. 
And when wooden buildings came into use, the wood was made to 
imitate stone in general form, sculpture, and painting. Hence it 
was that pillars became the elements of Architectui'e, whether of 
stone, brick, or wood. It has been suggested that the Gothic style 
grew out of the ancient practice among the Goths, Vandals, and 
other northern barbarians, of binding together the tops of slender 
trees, for covering with skins, bark, &c. That this gave rise to the 
sharp arch, and other peculiarities pertaining to that order. 

Sec. 154. The pillar consists of the hase, shaft, capital, architrave, 
frieze, and cornice. A column includes the base, shaft and capital. 
The pillar is extended laterally, or rather spread out, so as to con- 
stitute walls, bases, capitals, cornices, &c. And when the form of 
a column is assumed, its proportions are extended to all parts of a 
regularly constructed building. And this principle is rigidly applied 
to bridges, ornamented boats, carriages, rail-road cars, &c. 

Sec. 155. As pillars are the elements of the science of Architec- 
ture, they are to be first presented to the student. Figures or draw- 



86 

ings of the elementary pillars will greatly aid the student ; therefore 
he is referred to the various treatises on that subject. Benjamin's 
Practical House Carpenter is the most popular work in this country ; 
and, perhaps, quite as useful to the student as any other work. 
But the most efficient method of giving instruction on this subject is, 
to take students about a city or village ; and first, point out the va- 
rious orders of architecture by correctly constructed pillars — second, 
point out the errors ; which are always abundant — third, point out 
the lateral extension of pillars and parts of pillars, in the construction 
of walls, ceiling, door casings, window casings, cSzc. The teacher 
should call his pupil's attention most particularly to bridges, and 
other works, which do not come particularly under the daily opera- 
tions of common carpenters. 

THE FOUR PILLARS. 

Sec. 156. The four pillars are called Tuscan, Boric, Ionic, and 
Corinthian. The Composite order is of modern application. It is 
a fanciful intermixture of the elementary characteristics of some of 
the four orders — generally of the Ionic and Corinthian, This trea- 
tise being a mere practical text-book, will include the essentials of 
students' recitations, only. 

TUSCAN ORDER. 

Sec. 157. This order is the plainest and stoutest of all the orders. 
But it varies in proportion to the weight it is to sustain. In some of 
the cases of best appearance, the column is in height equal to seven 
times its diameter adjoining the base ; and the entablature is two 
diameters. Its parts may be thus enumerated : a square plinth be- 
low the base ; a base of a large moulding of a half cylinder ; a 
terete, or tapering cylindrical column ; a plain capital at the top of 
the column ; a plain architrave sitting somewhat towards the inside 
of the capital ; a frieze on the architrave ; and a broad over-laid 
coraice, projecting considerably forward — the entablature is thus 
made very plain. 

DORIC ORDER. 

Sec. 158. This is also a plain order, and resembles the Tuscan. 



87 

A little more slender than the Tuscan, and more ornamented. The 
columns are often fluted ; and the entablature and cornice are often 
ornamented with triglyphs and modlllions. 

IONIC ORDER. 

Sec. 159. This order is always distinguished by the volute or 
scroll. It is more slender than the Doric order, and is often highly 
ornamented with various kinds of sculpture. This order is more 
adopted in this country than any other, in all buildings of taste. The 
columns are mostly fluted ; and the architrave and frieze are more 
or less ornamented. The base is surrounded with more compli- 
cated mouldings than the Tuscan or Doric ; and the modillions are 
larger and generally of good workmanship. 

CORINTHIAN ORDER. 

Sec. 160. This order is distinguished by its plumose capital, and 
its more slender and delicate proportions. The base generally re- 
sembles that of the Ionic, and its column is generally fluted. Its 
architrave, freize, and cornice, are ornamented with sculptured 
work and curvelinear modillions. But its high elegant capital gives 
it a degree of grace and beauty, far exceeding the other orders. 

COMPOSITE ORDER. 

Sec. 161. This order, though compounded of two others, has its 
true characteristics. It has the plumose capital of the Corinthian 
order below, and the volute of the Ionic order above — but the volute 
is generally elliptical in a vertical direction. Its architrave and 
frieze are often highly ornamented. 

PEDESTALS. 

Sec 162. These are proportioned to their respective orders. 
They consist of hase, die (parallelepiped trunk,) and a cornice — 
altogether about a third as long as the column supported by them. 

PILASTER. 

Sec. 163. Pilaster is a pillar of any of the orders, which has the 
appearance of being partly sunken into the wall — or it may be de- 



88 

fined, as a pillar with a thin tapering parallelepiped column. It 
consists of a plank, or slab of free-stone, attached to a wall, fire- 
jambs, &c., constructed upon the principle of some of the orders. 

MISCELLANEOUS STRUCTURES. 

Sec. 164. Colonade, a series of columns, which ai-e united by en- 
tabliture at their tops. 

Arcade, several receding arches in succession, penetrating into, 
or through a building. 

Balustrade, a series of small pillars, as those supporting stair- 
railings, &c. A kind of parapet. 

Attic, often used for a garret ; because the pedestal-like pillars, 
which hide the roof (garret room) are called attics. 

Parapet or Battlement, any low wall or balustrade, surrounding 
a roof or covering of any works ; intended to conceal an unseemly 
part, or to defend it. Sometimes it surrounds roofs for the purpose 
of supporting the hand-railing of a walk. 

Pediment, generally a small triangular front roof, set in upon the 
slope side of a larger roof. 

Balcony, open gallery; as the iron galleries around upper win- 
dows, and highly elevated terraces. 

Belvidere, (beautiful view. French.) observatory, and turret. 

Cupola, a dome, or smallish room, on the top of a building ; as a 
belfry, or hemispheric sky-light. 

Terrace, elevated walk. 

Coping, top or binding-stone, or binding timber, on a wall. 

Saloon, a vaulted spacious hall. 

Corridor, a large hall or passage to distant apartments. Some- 
times applied to galleries or covered ways, leading around build- 
ings. 

Lintel, and Threshold, top and bottom pieces of a door opening — 
sometimes applied to the top, or covered part, of a projecting out- 
side room. 

Niche, appHed to recesses in walls, for setting ua urns, dz;c. 



89 



RAIL-ROADS, &c. 

Surveying a Route for a Rail-Road, MAdam Road, Turnpike 

Road, Canal, SfC. 

EXTEMPOKANEOTTS TrAVEESE. 

Sec. 165. If a road of any kind, or a canal, is proposed, across a 
mountainous, hilly, or even a moderately uneven country, a kind of 
extemporaneous traverse should first be taken. This will give the 
directors such a general profile, or rather view of the country, at a 
small expense, as may enable them to judge with considerable accu- 
racy, respecting the most expedient route for a preliminary survey. 
It may be conducted advantageously with a compass, chain, and 
barometer. 

Sec 166. Let the barometer be set up at the beginning of the 
line. Let the flag remain with the barometer, while the compass 
and chain are carried to the first considerable elevation. or depres- 
sion, within view of the flag. Here run a line, of a suitable length, 
between two stakes, for\he base line of two triangles — the apex of 
the first triangle to be at the first station of the barometer, and the 
apex of the other triangle to be at the next station of the barometer. 
The base line being carefully measured, and the angles at each end, 
formed by the line with the bearings of the barometer at each sta- 
tion, will be all that is necessary for taking two strides with suffi. 
cient accuracy. As the barometer will give a good approximation 
to the true height at every station, and as the distances may be 
pretty accurately taken ; a profile across an extensive district may 
be taken in a short time. From five to ten miles per day may be 
taken by an experienced surveyor, and a skilful barometer-bearer. 
If notes are extensively taken, much of the character of the country 
maybe presented on, or accompanying, the profile and sketch-book. 

Sec. 167. The barometer should be applied twice at every sta- 
tion in the same day. It is found, that when the atmosphere is uni- 
form, and the barometer is not influenced by change of temperature 
nor moisture, the barometer will sink the lowest at about 4 o'clock 
in both forenoon and afternoon ; and will rise the highest at 9 in the 
forenoon and 1 1 in the evening. This is supposed to be caused by an 
uniformly operating cause, above clouds or other modifications of 
aqueous vapor. Therefore the barometer ought to be twice set on 

12 



90 

the same day and place ; so that where it stands at 4 o'clock in the 
forenoon or afternoon, it should again stand at 9 in the forenoon or 
11 in the evening, and an average be made. But when this cannot 
be conveniently done, it may in some measure be approached, by 
assuming opposite times nearly in contrast ; as 9 A. M. and 4 P. M. 
But the medium hours do not need contrasting ; as 7 A. M. or IIP. 
M. But all cases require, that morning vapors should be exhaled 
before the barometer is used ; and that the season of the year should 
be chosen, when the weather is most dry and settled. 

Sec. 168. Formula for calculating Heights by the Barometer, ac- 
cording to Hulton. 

As the density of the atmosphere, consequently its weight, dimi- 
nishes in a geometrical ratio of its height, and as logarithms of num- 
bers are constructed upon the same principle, Hutton sought, with 
success, a formula for applying logarithms for taking heights of hills, 
mountains, &c., with the barometer. He found that, if the tempe- 
rature of the atmosphere stood at 31 degrees of Fahrenheit, the dif- 
ference in the first four figures of the logarithm, for every hundredth 
of an inch on the barometer between the bottom and top of a hill, 
gave just one fathom (6 feet.) Hence his rule: If the mercury in 
Fahrenheit's thermometer (alwa3's attached to the barometer) .stan^^s 
at 31°, take the height of the mercury in the barometer, in inches 
and hundredths of inches, at the top and bottom of the hill. Find 
the logarithm of both. Subtract the logarithm of the top from the 
logarithm of the bottom of the hill. The four first places of figures 
in the remainder are fathoms, and the remaining ones are decimals 
of fathoms. If the answer is required in feet, multiply the fathoms 
and decimals by 6, and the product will be the answer in feet and 
decimals of feet. 

Sec. 169. If the thermometer stands above 31°, an addition will 
be required ; to be produced by the following formula : Divide the 
fathoms and decimals of fathoms, by the constant number, 435; and 
multiply the quotient by the difference between 31° and the number 
of degrees on the thermometer. This product is to be added if the 
temperature is above 31°, but to be subtracted if below. 

Note. If there is any difference between the temperature at the 
top and bottom of the hill, the average is to be taken. 



91 



A portion of a table of logarithms, sufficient for our highest moun- 
tains, is inserted at the end of this treatise. 

Sec. 170. Gregory's formula has the advantage of being conve- 
nient for the memory in the absence of tables. A perpetual formula 
of 55 with three ciphers — 55 degrees of temperature as the stand- 
ard — 44 with a cipher, for a corrective for deviations from said 
standard. His rule is : Divide the difference between the top and 
hottoin hundredths of an inch, hy the sum of both heights; then multiply 
the quotient by 55000, and the product will be the height in feet. 
But if the temperature differs from 55°, add the 440th part of the 
height for every degree it exceeds 55°; and subtract the same for 
every degree below 55°. Example : 

Barometer. Thermometer. 

"Top of hill, 30.00 Average, 

Bottom, 29.80 68°— 55°= 13 



Difference, 


.20 


Sum, 


59.80 59.80).200000(.00334 




.00334 17940 




55000 formula. 



1670000 
1670 



440)183.70000(.04175 
1760 13 



20600 
17940 

26600 
23920 

2680 



770 
440 

3300 
3080 



.12525 
.4175 

.54275 
183.70000 



2200 184.24+Answer. 
2200 



92 



Latitude and Longitude Surveys. 

Sec. 171. Very extensive RaiJ-roads (like that now in progress 
from Lake Erie to New-York,) or Canals (as that from Lake Erie 
to Hudson river,) should have all remarkable points, along the va- 
rious proposed routes, accurately settled by their latitude and longi- 
tude. This would greatly aid the judgement of directors; and 
greatly benefit large districts of country, by furnishing established 
points for future reference. 

Sec. 172. Latitude is most conveniently taken by the sextant, at 
noon. But Longitude ought to be taken, inland, in most cases, by the 
eclipses of Jupiter's satellites. I will, however, describe the me- 
thods of taking longitude by Jwpiter^s eclipses and by three-hour 
lunar ohservaiions. 

Sec. 173. Taking latitude at noon with the sextant, requires a 
nautical almanac; though some of the larger almanacs of the 
common kind, contain the sun's declination, and may be used as a 
substitute. But no engineer should fail to provide himself with the 
Annual Nautical Almanac ; always published three years ahead, 
by the Messrs. Blunts, of New-York. 

Note. Mr. Gates, of Troy, will furnish them to order. 

Sec. 174. In taking the latitude at noon, a reflector is necessary. 
Bowditch prefers a bowl of molasses to a glass reflector. Set a 
bowl of molasses (a large soup-plate is preferable,) on the ground, 
and take the angle between thw sun and its reflected image in the 
molasses, by bringing them centre to centre. This gives the double 
altitude ; as the distance of the sun's image, below the level of the 
surface of the molasses, is equal to that of the real sun above it. 
Halve this double altitude, which gives the true altitude. 

Sec 175. Having obtained the sun's true altitude, proceed to cal- 
culate the latitude, as follows : Look out the sun's declination — If 
the declination is north, (as it must be, from the 21st of March to 
the 21st of September,) subtract it from the altitude ; and then sub- 
tract that remainder from 90 degrees, which will leave the latitude. 
If the declination is south, (as it must be, from the 21st of September 
to the 21st of March,) add it to the altitude, and subtract the sum 
from 90 degrees, which will leave the latitude. In short days, wher^ 



93 

tthe sun runs low, an allowance may be made for refraction, accord- 
ing to the table of refraction at the end of this treatise. 

Note. In the longest days of summer, the sun will be too high 
at noon to admit of double altitude within the range of the sextant. 
Several methods are in use for obviating this difficulty. The fol- 
lowing method may be adopted : 1st. Take the double altitude of 
the ridge of a house-roof, or some other straight horizontal line. 
Then wait for noon, and take the single altitude of the sun above 
said ridge, &c., and add it to half the double altitude of said ridge. 

Sec. 176. In taking the lojigitude by the eclipses of Jupiter^s 
satellites, no instruments are necessary but a telescope and a good 
time-piece. On land this method of taking longitude is the best. 
Proceed as follows : Look in the Nautical Almanac, in the monthly 
table of Jupiter's satellites, and find the time of the nearest immer- 
sion or emersion eclipse. Be prepared with the true time and teles- 
cope. Direct the telescope to Jupiter, with the slide drawn so as to 
give its largest size, a few minutes before the time of the eclipse. 
With the eye on Jupiter, move the slide so as to diminish it, until the 
satellites come within the field of vision. Then wait until the im- 
mersion or emersion occurs. If immersion is to occur, expect its 
disappearance a little before an apparent contact. Both immersion 
and emersion will appear suddenly. Note the instant of its occur- 
rence, by the waitch. Then calculate the difference in time between 
its occurence and the time set in the Nautical Almanac. Allow 
15 degrees of longitude for every hour, and the same proportion for 
minutes and seconds, and you have the degrees and minutes of long- 
itude from Greenwich at London. 

Sec. 177. In taking the longitude by lunar oiservations, a good 
sextant, or reflecting quadrant, and a good time-piece, are neces- 
sary. Look into the Nautical Almanack, and find the angular dis- 
tance between the moon and the sun or aplanet, or one of the nine fix- 
ed stars, which are used for this purpose, which may be seen at the 
time of night or day required. These stars have been selected so 
as best to accommodate every part of the earth, and to be of suffi- 
cient magnitude for observation. They are called a (alpha) of 
Arietes — a (alpha) of Aldeiaran — Pollux — Regulus of first and 
second magnitude — Spica, first magnitude — Antares — AquilcB re- 
markably bright — Formalhaut, smaW^Pegasus, a and h. By ex- 



94 

amining these stars on a celestial globe, or map, particularly Burrit's 
Atlas of the Heavens, the student may soon make himself sufficiently 
familiar with their relative positions, to find them at one glance of 
the eye. In taking lunar observations, half an hour of shewing is 
better than ten days of reading. The sextant must be set according 
to the Nautical Almanac, for the nearest third hour. As the time 
approaches, look at the moon and sweep for a star ; but look at the 
star and sweep for the moon as they approach each other. Note 
the instant they touch, according to the almanac and watch. 
Then calculate the longitude by allowing 15 degrees for every 
hour's difference between your time and the time given for Green- 
wich at London. Students who have no experienced teacher near, 
must read the directions given by Bowdiich. 

Hours of the day are reckoned from noon to noon ; counting from 
noon, to 23 o'clock and 59 minutes. Parallax and refraction must 
be allowed according to the tables at the end of this treatise. 

Sec 178. Should a surveyor be called to lay offa piece of ground 
of great extent, (as the Oblong, taken from Connecticut last cen- 
tury and joined to New-York,) which was to be a true north and 
south parallelogram, he would be under the necessity of calculating 
the breadth of a degree of longitude at the north and south ends of 
the tract, and projecting it upon Mercator's method. To find the 
breadth of a degree of longitude at any degree of latitude, state thus : 
As radius at the equator is to 69.1 miles, so is the co-sine of the 
degree of latitude to the measure of a degree of longitude at the 
given degree of latitude. 

Nat. sine of 90®. Miles. Nat. co-sine of 40*. 

Asl.pOOOO : 69.1 :: .76604 

.76604 



2764 
41460 
4146 

4837 

52.933364 



Answer, 52.93 miles. 



95 

RAIL.ROAD SURVEYING.* 

Preliminary Survey. 

Sec. 179. A full party for the field operations of a preliminary 
survey is composed as follows : 
One Chief Assistant Engineer, 

" Compass-man or Surveyor, 

" Assistant do. 

" Leveller and Assistant Leveller, 

" Rod-man, 
Two Chainmen, 

" Ax-men, 
One Flag-man. 

Sec. 180, With these the chief assistant goes into the field. He 
is supposed, of course, to have been previously made acquainted, by 
the chief engineer, with the general direction of the proposed rail- 
road, and some of the principal intermediate points through which 
the line is expected to pass. 

Sec 181. The principal objects of a preliminary survey are, to 
ascertain the distance between any given points, the difference of 
elevation between those points, and also the intermediate ground 
traversed, together with a general sketch and description of the dif- 
ferent lines pursued in reaching the desired place. 

Sec. 182. The distance and difference of elevation of any two 
points are necessary to enable the engineer to determine whether 
the rise or fall per mile, or the grade, as it is technically called, can 
be such as will render the motive power proposed effective. The 
beai'ing of the lines, together with the topographical sketches and 
field notes, ai'e indispensable in determining the quantity or degree 
of the curvature. 

Sec 183. After being made acquainted with the general direc- 
tion of the line, the chief assistant traverses the ground, examines it 
carefully for some distance in advance, and having determined upon 
the ground upon which he will run, directs a flag to some point as 
far from the place of beginning as it can be distinctly seen, and the 

* This arlicle was obligingly furnished by Engineer Surgent. 



96 

assistant at the compass takes the courses as in ordinary land sur- 
veying, laying off the line in stations df two hundred feet each, by 
means of stakes driven firmly in the ground and numbered, making 
No. 1, 200 feet from the point of commencement, and so on. During 
this process the chief assistant carefully sketches the topography upon 
each side of his line, directs bearings to be taken to the most pro- 
minent objects in the vicinity, notes the character of the soil, spring 
runs, or streams of water that cross the line, and determines the 
size of a sluice or drain necessary to pass the water. This done, 
the instrument is moved forward to the point where the flag stood, 
or some station in line with it, and the course tested by a back sight 
along the line of stakes to the point of commencement ; when the 
same process is repeated, unless from the formation of the ground it 
is necessary to change the course. If this is the case, an angle is 
made of such magnitude as may be directed by the principal officer 
in the field ; having due regard to the effect that will be given by 
tracing a circle between the two lines of which they will be tan- 
gents. 

Sec. 184. The level follows the compass, using some point near 
the commencement as a base, and taking the relative heights of each 
station. This is usually performed by setting the level at station 
No. 2, and directing the rod-man to hold his target on the point es- 
tablished as a base, and to move the vane as directed until the level- 
ler exclaims,jras!5. The rod-man makes fast and re'plles fast ; when 
the leveller again looks, and if tiie horizontal hair of the instrument 
corresponds precisely with the middle of the vane, calls right. The 
rod-man then carefully reads from the rod the feet and decimals 
above the surface, and calls the result in a quick, sharp but distinct 
voice. This the leveller, assistant leveller, and rod-man, enter in 
their books, under the column of B-sights, and the rod-man moves 
forward to the next station and holds up on the surface, observing 
that he gets the natural range of the ground ; when the observation 
is repeated as before, the result called, and entries made under the 
head of Fore-sights. Each then takes the difference and places it 
under the head Above, if the back sight is greatest, and under Be- 
low, if the fore-sight is greatest. The leveller calls the result, and 
the assistant and target-man assent or dissent, as their results agree or 
disagree with his. This done, the last Fore-sight is brought down 



97 

and placed under the head of Back sight, and an observation taken 
to the next station, the result of which is placed under the head of 
Fore sight, and the difference again taken ; and if the back sight is 
greatest, added to the last result, if less deducted. Again the last 
fore sight is brought down, and the same process repeated, giving 
the result for station No. 3 ; and the rod-man goes to No. 4, and 
directs a small pin to be driven, about six inches from the stake, 
firmly and close to the ground, upon which he places his rod for the 
observation, which when taken finishes the business of the "set," 
and the leveller moves forward, takes his station at No. 6, and re- 
peats the process before described ; unless, as is frequently the case, 
the undulations of the earth prevents his getting his sights from the 
regular stations. When this is the case, he avails himself of the 
most favorable position, having command of the most stations, and 
giving equal distances between the 2^sg driven and the one upon 
which he again proposes to shift his level. The annexed table 
shows the manner of entering the field notes. 





Dis- 


Back 


Fore 


Differ. 








No 


tance. 


Sig'it. 


Sight. 


ence. 


Above. 


Below. 


Remarks. 


1 


2U0 


8.204 


3.30 


+4.904 


4.90 




Start on the sur- 


9, 


« 


3.30 


4.70 


—1.40 


3.50 




face of the rail at the 


3 


a 


4.70 


2.60 


-{-2.10 


5.60 




west end of Troy 

Bridge. 

-Peg. 


4 


ii 


2.60 


1.406 


1-1.194 


6.794 




5 


a 


9.465 


6.54 


+ 2.925 9.719 






6 


a 


6.54 


4.75 


+ 1.79 


11.509 






7 




4.75 5.302 


—0.552 


10.957 




Bench No.l on hick- 
orr tree west of line. 



Note. It will be observed that the instrument has been changed; 
but the same process is necessary as in the preceding case, and the 
only difference is the relative position of the peg to the base first 
started with. The level from elevated positions is usually turned 
in various directions, to render it certain, the eye of the engineer 
has not been deceived in selecting the general route. 

Sec 185. After the field operations cease for the day, the level- 
ler will examine his notes, comparing them with his assistant and 
rod-man, and foot his back sights and fore sights, to see if the differ- 
ence correspond with that obtained in the field, also add or deduct, 

13 



98 

as the case may require, to or from the starting point, and compare 
the result with that obtained at the point of leaving off. 

SeCo 186. When a line for a rail-road has been traced as above 
described, the second step is to make a rough, or as it is termed, a 
working profile, and the engineer proceeds to adapt thereto the best 
grades it will admit of, and from the notes of the chief assistant, le- 
veller, and compass-man, together with his personal observations, 
to suggest such changes, modifications, and improvements, as his 
judgment dictates, for the guidance of his assistant in executing the 
^^ Definitive Survey," which is the next step preliminary to breaking 
ground. 

Sec. 187. The level in this follows the compass, as in the preli- 
minary survey, noting every material deviation in the surface over 
which the line is traced ; also, the level at the surface of water in 
all streams that are crossed, together with the soundings, and esta- 
blishing frequent benches or permanent levels, on stumps, trees, or 
rocks, adjacent to, and convenient for the future adjustment of the 
line. The stations are now reduced to 100 feet, Euad when it is 
necessary to take intermediate levels, which is frequently the case, 
they are entered in the field notes thus ; 



No. 


Dis. 
tance. 


Back 

Sight. 


Fore 

Sight. 


Differ- 
ence. 


Above. 


Below. 


Remarks. 


40 
41 


100 
30 
40 
30 















Sec. 188. The duties of the level thus completed in the field, a 
second, mor€! accurate and finished, profile is made, the grades 
adapted to it with the utmost care — the streams represented, also 
the division lines of farms, by a small flag or spear, and the name 
of the owner of each separate lot or farm, neatly printed between 
the boundaries, together with other general remarks, such as " op- 
posite Bethlehem church," " road to the Shakers," " steam saw 
mill," &c. &c. Also the ratio of the grade per 100 feet and mile. 
The profile being thus far complete, and the grades satisfactorily 
arranged, the cuttings and fillings are next to be made out, and placed 



99 

along a line, drawn parallel to the base line of the profile — ^the cut- 
tings being placed above, and the fillings below. 

Sec. 189. The cuttings and fillings are deduced from the levels 
and grades in the following manner : 



No. 


Dis- 
tance. 


Surface. 


Grade. Cutting. 


Filling. 


470 
71 

72 


100 


246.57 
246.71 
260.00 


•25U.87 
251.71 
252.55 7.45 


4.30 
5.00 



The grade being started at the base with the surface, is readily 
calculated from the rate which has been previously established. 
When the undulations of the ground are very abrupt, the interme- 
diates are sometimes deduced in this survey, but not generally until 
the next 



Staking Out. 

Sec. 190. This consists in laying off" the work and defining its 
boundaries, so as at the same time to procure the necessary notes 
for a correct measurement of the quantity of earth to be removed or 
supplied, and direct the contractor how to proceed with the execu- 
tion of his work. 

Sec. 191. The slopes necessary to be given in excavations and 
embankments, are determined from the nature of the soil. In em- 
bankments however, it is not usual (as we say,) to make ihem less 
than 1 J to 1 ; that is, with a 1^ base to 1 rise or perpendicular. 
These, however, vary greatly according to the views of different 
engineers, as well as from the circumstances above stated. We 
will suppose, then, that it is determined to lay off the road for a 15 
feet width of bed on the top. Going, then, to station No. 10, we find 
it marked and also entered in our grade book — 4.00 (4 feet below.) 
The instrument is placed and the rod sent to the nearest bench, 
which was marked +2.00 — S. 10, (meaning two feet above grade 
at station No. 10.) The instrument was placed so as to coiTimand 
a view of as many stations as convenient, and the sight taken, which 
was 6.00. This was added, on a bit of loose paper, to the 2 feet, 



100 

making 8, and the letter T placed next the result, understood tripod* 
or that the cross hairs of the level were 8 feet above grade at sta- 
tioil. The leveller then moved up his instrument to No. 10, and 
casting his eye to the right of the line, judged that the ground rose 
about 1 foot in 12, and gave the target-man the ring of the tape, 
and directing him to move off at right angles to the line, entered 
into the following calculations in his head : (Centre 4 feet. — 1 foot 
rise leaves 3.— 1+3=4-|- and 4-|- +71 = 12.) At 12 feet then, the 
rod is held up and sight taken, which proves to be 12 feet. The 
tripod deducted showed the ground to be 4 feet below grade, and 
that he had not found the medium and was too near the centre, as 
running in his ihind again the same calculation, he discovered that 
4 feet fill would require HH feet distance from the centre ; so the rod 
was again held up at IS^ feet, and the hair of the instrument cut the 
target in the centre, which had not been moved, hence gave the 
proper point of intersection with the surfoce of the proposed line of 
slope. The calculations previously gone through with in the head, 
were now made on paper by all the assistants comparing and agree- 
ing ; it was entered in the field book as shown in sec. 000. The op- 
posite or left stake was set ofFin like manner, and the leveller and tar- 
get-man moved on to station No. 11, and the grade ascending j-*/^ 
of a foot in a hundred, 0.40 was taken from the tripod, which gave 
7.40 as the tripod for No. 11. 

CURVES.* 

Sec. 192. One of the most difficult parts of the field operations 
of a locating survey, and that perhaps in which more skill and 
judgment is necessary, than in any other particular case, is the 
changing the direction by curving, in a broken or hilly country; 
when attention must, at the same time, be paid to keeping upon a 
given level, or maintaining a uniform ascent or descent. Two 
methods are in common use for tracing curves. One by suc- 
cessive deflections with the compass or theodolite ; the other 
by measuring offsets (secants) from tangent lines. The use 
of the compass is sufficiently explained in sec. 110, 111, and other 
parts of land surveying. If a theodolite is used, it should be of 

* This article was prepared by engineer C. B. Evar.s. 



101 

the best make and graduated with the greatest possible care. If 
such is the case, a curve may be traced with a great degree of ac- 
curacy, in the following manner : Place the instrument on the sta- 
tion at which it is proposed to commence the curve, see that the zero 
of the nonius precisely corresponds with the zero of the graduated 
limb of the instrument, turn the mstrument until the vertical hair 
exactly cuts tlie row of stakes and flag at the opposite end of the 
line, then tighten the clamp screw to secure the lower limb from 
moving, let down the needle and note the course of the line. You 
are supposed to be running on or near the line previously run in the 
preliminary^ survey, and therefore know about the number of de- 
grees contained in the angle at which you are about to trace your 
curve. From this the quantity or degree of the curve is determined, 
and the curve, in technical parlance, is named from the number of 
degrees deflected from the course at each station, thus : if at each 
station the course is changed in the same direction two degrees, we 
call it a two degree curve ; if the course is changed three degrees, 
a three degree curve, and so on. Flaving determined what the de- 
gree of curve shall be, loosen the screw, which connects the move- 
able and stationary plates of the instrument, and turn the moveable 
part in the direction you wish to curve until the zero of the nonius 
corresponds with half the number of degrees on the graduated plate 
that you have determined upon as the quantity of the curve, thus: 
If you propose to trace a two degree curve, after setting the instru- 
ment upon the line as before, you will turn it round until the zero 
of the nonius cuts 1, or the first division on the card. Then let the 
chain be straightened from the station at the instrument, and a stake 
driven in line with the instrument as last set. This will be the first 
station in the curve. Then move the instrument until zero of nonius 
cuts 2° on the card, or one degree further than when the last stake 
was set, and at the end of another chain set another station, and for 
each station that you set while the instrument remains where you 
first placed it, you turn it one degree on the card. In this way you 
may set 8 or 10 stations in the curve without moving up with the 
instrument. But it must be borne in mind, that the farther you pro- 
ceed from the instrument the more likely are you to vary from a 
true curve ; therefore, it is generally advisable not to set more than 
8 or 10 stations without moving up the instrument. After setting 



102 

the stations as above, as far as can be seen with accuracy, move up 
the instrument and place it precisely over the centre of the last sta- 
tion set. Turn the instrument back again to zero, direct it to the 
first station back, and let down the needle to test the course. While 
the needle is settling, you may calculate what the course ought to 
be, as follows : Add together the number of degrees deflected in the 
curve, and add the amount if curving from the nearest magnetic 
pole, or subtract the amount i? cnw'mgtowards it, thus: The course 
of a line was N 20° VV, and a curve 2° to the right, was commenced 
and run with the instrument nine stations, when it was necessary to 
move up. 

Sec. 193. The instrument was carried to the last station and set 
as above directed. While the needle was settling the course was 
calculated thus : At the first station in the curve the deflection from 
the tangent was 1 degree, and the 8 following stations were 2 de- 
grees each, making in all 17 from the course of the tangent. The 
bearing of the tangent was N 20° W, and the curve towards the 
north or the nearest magnetic pole ; tlierefore, subtract 17°, the 
number deflected from 20 the course of the tangent, and it leaves 3° 
or N 3° W as the bearing of the last chord. On looking at the 
needle, the course as then indicated was found to correspond with 
that calculated, and therefore the work was known to be right. 

Sec. 194. After directing the instrument to the last stake, tighten 
the clamp screw and turn the instrument 2° on the card for the next 
station ; but for every station after that deflect but 1°, until the in- 
strument is again moved up. When the course is . sufficiently 
changed to stop curving, the instrument must be moved up and 
placed on the station which you propose to make the end of the 
curve. After directing the instrument to the last stake as before, 
deflect 1° on the card, and the theodolite will then be right for run- 
ning a tangent to the curve. Set the stakes on the tangent as far 
as you can distinctly see ; but before you move the instrument, test 
the course and see if the needle corresponds with what you make it 
by calculation. 

Sec. 195. Running curves by offsets from tangents, depends 
upon the same principle as the one above described ; but instead 
of deflecting with the instrument as above, departure equivalent 
to the degrees which would be deflected, is measured off from 



* 103 

the tangent or the chord of the last station, produced as the case 
may be, thus : If you wish to trace a 2° curve by measurement, 
you will produce the tangent one station farther than the one at 
which you wish to commence the curve; then, with a tape line 
graduated to decimals of a foot, measure off the departure for 1° ' 
straighten the chain from the last stake to the end of the tape, and 
then set your stake. Then produce the line of the last two stakes 
a hundred feet farther, and from that point measure off the depar- 
ture for 2° ; bring the end of the tape to the end of the chain as 
before, drive your stake, and proceed on in this way to the end of 
the curve. 

Compound Curves. 

Sec. 196. The above methods describe a simple or regular curve, 
or a section of the circumference of a circle. It is sometimes neces- 
sary, after proceeding some distance with a curve, to change the 
degree of it, and curve faster, or not so fast, as the case may be • 
that IS, a greater number of degrees are deflected at each station^ 
or the curve made to conform to a shorter radius, thus: if after 
runnmg some distance on a two degree curve, you find it necessary 
(to suit the formation of the ground, or from other causes,) to change 
your curve to 3°, you will proceed as follows : At the station where 
you wish to change make a deflection of two and a half degrees 
and at the next three, and so on, as far as you continue the curve' 
In every case where you change from one degree of curve to ano- 
ther, add half the difference between the curve you are running and 
the one you wish to run, to the least, for the deflection at the point 
of change. 

Reverse Curves. 

Sec. 197. It sometimes becomes necessary to change the direc 
tionofthe curve altogether, and without the intervention of a tan 
gent, to change immediately from a curve in one direction, to one of 
a duecfon precisely opposite. When this happens, the operation is 
as follows: At the point where you wish to change, produce the 
chord as usual, but do not deflect; drive three stakes in line, and 
then commence deflecting in the opposite direction. 



104 

Pencilling and Calculating Cxjkves, 

Founded upon long Traverses through hilly Districts. 

Sec. 198. After plotting an extensive traverse, taken for a road, 
and sketching some of the most important objects minuted in the 
laeld notes, proceed to pencil out the proposed road, so as to suit the 
eye or fancy. Then divide off the pencilled road into arcs; each 
arc extending as far as the curviture continues to be uniform. Then 
draw a chord line to each arc, and fix the point at each end by mea- 
suring from the nearest points in the surveyed traverse, if the ends 
do not fall upon any surveyed angle. As there will, probably, 
always happen an angle at some point in the arc, find the length 
of the chord line by the case in trigonometry, where two sides and 
a contained angle are given. 

Sec. 199. Consider the angle as moved to the middle of the arc ; 
for the angle will be the same in any part of the curve, according to 
a known principle in geometry. Double this angle and subtract 
that sum from 360 ; and the remainder will be the angle at the 
centre of the circle of the proposed arc, made by the radii limiting 
it. Connect these angles by a diagonal line (halving both of said 
angles,) and this will make four right angled triangles, each with a 
known horizontal leg ; it being half of the long chord line. Find 
the length of the radius, as the hypothenuse, in the common manner 
of proceeding with similar triangles. 

Sec 200. Having found the radius of the arc, and the length in 
degrees, (which is the said angle at the centre,) find the measure of 
the arc in feet thus : Double the radius (making the diameter of the 
whole circle) and multiply the sum by the formula 3.1416— this 
gives the measure of the periphery of the whole circle. Then say, 
as 360° to the whole measure of the periphery ; so the degrees of 
the arc (as expressed by the angle at the centre) to the measure of 
the arc in feet. 

Sec 201. The foot measure, and the degrees of the arc being 
known, divide the foot measure of the periphery by 100 feet, and 
the degrees by that quotient. This will give the number of stakes 
to be set, and the degrees of each isosceles triangle at its apex in the 
centre, for each portion of the periphery of the arc staked out. 



105 

Sec. 202. You are now ready to stake out the periphery into 
hundred feet portions (the usual practice.) Although these are 
chord lines, unless the curvature is too great for any rail-road or 
canal, such short chords will coincide so nearly with the curve, that 
they will come out about equal— or an allowance may be made by 
shortening the chain a few inches. 

Sec. 203. The degiees for inflexion (or deflexion from the tan- 
gent,) at every stake, is to be conducted as follows: Find the direc- 
tion of the radius, and set the compass on it. Then turn the com- 
pass around 90°, which brings it upon the tangent line. (The tan- 
gent hne being always at right angles with radius.) Then deflex 
from the tangent line equal to half the angle at the centre of the 
cu-cle, which is made by the radii limiting the 100 fefet chord. But 
at every following stake, deflex equal to the whole angle at the 
centre ; from the last line run. 

Sec 204. Fix the said tangent line as follows : The chord line 
will form an angle with a line in the traverse ; from which the said 
chord hne was calculated. This angle can be found in the com- 
mon way for finding similar angles. Then sight the compass on 
said traverse line, and turn it through the number of degrees re- 
quired for bringing the compass upon the chord line. Lastly, turn 
the compass through the number of degrees found by calculation 
between the chord line and radius. This brings the compass upon 
radius, as required. 

Sec 205. Several other methods are in use among engineers. 
One is, to find the chord line of half the arc, and the versed sine, by 
trigonometry, instead of finding the radius as before explained. 
Then say : As the versed sine is to the said chord line ; so is the 
said chord line to the diameter of the circle. Then proceed with 
the formula 3.1416, as before directed. Also halve the diameter 
to obtain the radius, to be applied as before. 

Sec. 206. After having the general chord line, the chord line of 
half the arc, the radius, and angle at the centre, the periphery of 
the curve may be staked out by measures on the general chord line, 
and by ofl:sets, as follows : The general chord line and the radii 
meeting each end of it, constitute a general isosceles triangle ; con- 
sequently the base angles are known, according to division 4, of 
section 32. Each staked measure and the radii meeting each end 

14 



106 

of it, constitute an isosceles triangle also, with larger known angles 
at the base. Subtract a base angle of the former from a base angle 
of the latter, and the remainder will be one of the acute angles in 
the right angled triangle, made up of a staked measured line as hy- 
pothenuse, and the oiFset, and run lines on the general chord line 
as legs ; which two last lines can be found as in all cases of right 
angled triangles. After the first stake, the staked measured line 
will not form the hypothenuse for finding the offsets and run lines. 
But it must be found by doubling the first angle at the centre, and 
taking a radius for the middle term in the rule of three. In other 
respects proceed as before. Continue thus to calculate the run and 
off*set to the middle of the arc ; and apply the same measures for the 
other half. 

Sec. 207. A curve may be staked out by running all the lines 
from one end, thus : The first staked measure begins at one end, of 
course. Find the next chord line from the end, as in the run and 
offset method, before described. All the chord lines may then be 
found in this proportion : As any one of the chord lines is to the 
sum of the two adjoining, so is any other chord line to the sum of 
the two adjoining. Substitute cypher for the outside chord line of 
the first measured line, for the pui-pose of uniformity, and proceed 
thus : Suppose P, at the point of beginning. Suppose Q R S T V 
W, at the terminations of all the chord lines, where the stakes are 
to be set in the periphery of the arc. Then say : 

PQ : PO+PR :: PR : PQ+PS 
3.90 : 0+7.40 : : 7.40 : 14.04 

PS 
Subtract 3.90 from 14.04=10.14 

PR : PS+PQ :: PS : PR+PT 

, 7.40 : 10.14+3.90=14.04 : : 10.50 : 19.23 

2d.( 

PT 

Subtract 7.40 from 19.23=11.83 




107 



3d.^ 



PS : 


PT+PR :: PT : PS-f-PV 


10.14 ; 


; 11.83+'7.40=19.23 :: 11.83: 22.43 




PV 




Subtract 10.14 from 22.43=12.29 


PT 


: PS+PV : : PV : PT+PW 


1.83 


: 10.14 + 12.29=22.43 : : 12.29 : 23.30 




PW 




Subtract 11.83 from 23.30=11.47 



4th. 



The radius had previously been found to be 6.18. 

Having calculated all the distances from the station at the end of 
the curve, the courses only are left to be found. Fix on the tangent 
Ime as directed in sec. 204. Run the first measure so as to form an 
angle i.viih the tangent, equal to half an angle at the centre of the 
circle of the curve, made by two radii limiting said measured line. 
In this example the angle is , half angle. In running all the 
other lines, deflect half the said angle (as vi^ith the first measure,) 
from the last preceding course run. This will bring all the stakes 
to their true places in the periphery of the curve. 

Sec. 208. The method of running on the chord line and making 
offsets, or ordinates, to places for setting the stakes in the curve, de- 
scribed in the last preceding section, may be calculated from these 
chord lines, PQ, PR, &c. Call each of these lines the hypothenuse 
of a right angled triangle, and suppose a vertical leg let fall upon the 
chord line of the whole arc, and you then have the acute angle at 
P; of course the run and offset (ordinate) are found. 

Sec. 209. In running principal, or primary curves, the first me- 
thod proposed (runnmg on the periphery by inflections towards the 
centre — perhaps rather deflexions from the tangent,) is in general 
use. But in staking out the sub-curves, offsets, usually called ordi- 
nates, are chiefly used. The chord line of a sub-curve is now made, 
by most practising engineers, just one hundred feet in length. This 
is subdivided into portions of five feet each. Ordinates are thence 
set off" to the places for the exact location of the curve. These may 
be calculated by one of the preceding rules, or by sec. 211 to 215. 



108 

Sec. 210. The Ordinates for staking out the sub-curves, being 
very numerous, the calculations are tedious. It is on this account 
that a 100 feet chord is assumed, as a common length for the chord 
of a sub-curve, and 5 feet as a common length for the distance be- 
tween ordinates. This enables the engineer to calculate a table of 
-ordinates, to fit all cases ; and thereby to save much labor. In 
addition to this advantage, he avoids perpetual errors, which might 
be committed by less accurate assistants. 

Table of Ordinates. 

This table runs no farther than to the middle offset of the sub- 
curve, (usually called the versed sine,) because by inverting the 
order of the ordinates, the other half may be similarly staked out. 
The calculations for the lengths of the ordinates, are made to every 
degree and half degree of the first deflection from the tangent, from 
1° to 14°. This deflexion is always equal to half the angle at the 
centre of the circle of the curve, made by the meeting of the two sup- 
posed radii, limiting a sub-curve. 



109 



TABLE OF ORDINATES. 



05 

s 


Pri. 

mary 

An- 

gle of 

Def. 


6 

'V 
O 


6 

t-i 

o 

Ft. 


CO 

6 
O 


d 

"P 
O 


d 

o 


d 
O 


d 
O 


00 

d 
O 


d 
O 




D.|M. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


1 


14 


30 


.60 


1.12 


1.61 


2.02 


2.37 


2.65 


2.87 


3.03 


3.13 


3.16 


2 


14 




.58 


1.10 


1.56 


1.96 


2.25 


2.57 

2.48 


2.78 


2.93 


3.00 


3.05 


3 


13 


30| 


.56 


1.06 


1.50 


1.89 


2.21 


2.68 


2.83 


2.89 


2.94 


4 


13 




.54 


1.02 


1.45 


1.82 


2.13 


2.38 


2.58 


2.73 


2.78 


2.84 


5 
~6 


12 
12 


30 


.52 


.98 


1.39 


1.75 


2.05 


2.29 


2.48 


2.62 


2.68 


2.72 




.50 


.94 


1.34 


1.68 


1.97 


2.20 


2.38 


2.52 


2.57 


2.62 


7 


11 


30 


.48 


.90 


1.28 


1.61 


1.88 
1.80 


2.11 

2.02 


2.28 


2.41 


2.47 


2.51 
2.40 


8 


11 




.46 


.86 


1.22 


1.54 


2.18 


2.31 


2.36 


9 


10 


30 


.44 


.82 


1.17 


1.47 


1.72 


1.93 


2.08 


2.20 


2.26 


2.29 


10 


10 




.42 


.79 


1.11 


1.40 


1.64 


1.83 


1.98 


2.10 


2.15 


2.19 


11 
12 


9 
"9 


30 


.40 


.75 


1.06 


1.30 


1.56 


1.74 


1.88 


1.99 


2.05 


2.08 


.37 


.71 


1.00 


1.26 


1.47 


1.65 


1.78 


1.89 


1.95 


1.97 


13 
14 

Is 


8 
"8 


30 


.35 


.67 


„94 


1.20 


1.39 


1.56 


1.69 


1.78 


1.84 


1.86 




.33 


..63 


.89 


1.13 


1.31 


1.47 


1.59 


1.68 


1.74 


1.75 


7 


30 .31 


.59 


.83 


1.05 


1.23 


1.38 


1.49 


1.57 


1.63 


1.64 


16 


7 


.29 


.55 


.78 


.98 


1.15 


1.28 


1.39 


1.47 


1.52 


1.53 


17 

18 
l9 


6 
~6 
~5 


30 


.27 


.51 


.72 


.91 


1.07 


1.19 


1.29 


1.37 


1.41 


1.42 


30 


.25 
.23 


.47 
.43 


.67 


.84 


.99 


1.10 


1.19 


1.26 


1.30 


1.31 


.61 


.77 


.91 


1.01 


1.09 


1.16 


1.19 


1.20 


20 
21 


5 

~4 


30 


.21 


.40 


.56 


.70 


.82 

.74 


.92 


.99 


1.05 


1.08 


1.09 


1 .19 


.36 


.50 


.63 


.83 


.89 


.95 


.97 


.98 


22 


4 




.17 


.32 


.44 


.56 


.66 


.73 


.79 


.84 


.86 


.87 
.77 


23 


3 


30 


.15 


.28 


.39 


.49 


.57 


.64 


.69 


.74 


.76 


24 

25 


3 

~2 




.13 


.24 


.33 


.42 


.49 


.55 


.59 


.63 


.65 


.66 


30 .11 


.20 


.28 


.35 


.41 


.46 


.50 


.52 


..54 


.55 


j2e 


2 


.08 


.14 


.22 


.28 


.33 


.37 


.40 


.41 


.43 


.44 

.33 
T22 


2' 


^ 1 


30 .06 


.10 


.16 


.21 


.25 


.28 


.30 


.31 


.32 


2^ 


5 1 


.04 


.07 


.11 


.14 


.16 


.18 


.20 


.21 


.21 



110 

Sec. 211. To understand the table of ordinates, you should learn 
the method of calculating it. Proceed as follows : To have a clear 
view of the subject, plot an isosceles triangle. Make by any scale, 
tne oase 100 feet according to the practice of civil engineers in this 
country — this being a chord Hne of an assumed, or given, curve. 
Make the two equal sides at random ; but at least three or four 
hundred feet. As the degrees of deflection of this chord, from tan- 
gent, must be given, [see sec. 206] double this and you have the 
angle of the apex of the isosceles triangle ; which is the angle form- 
ed by the two limiting radii of the curve, at the centre of the circle, 
of which it is an arc. Then by trigonometry say : As the angle 
at the apex is to the hundred feet base line ; so is half the remain- 
der, after subtracting the angle at the apex from 180°, to the length 
of the radius. 

Sec. 212. Double the radius gives the diameter of the circle of 
which the curve is an arc. Then add 50 feet (half the given chord 
line of the curve,) to half the diameter. Subtract that sum from 
the whole diameter ; then multiply that sum by the remainder, and 
extract the square root of the product. This will give a standing 
ordinate, to be subtracted from all future ordinates to be obtained 
as hereafter directed ; which remainders will be respective ordinates 
of the above table. 

Sec. 213. The mathematical principle on which these calcula- 
tions depend, is this : Wherever the diameter of a circle may be 
cut, the two parts being multiplied together, and the square root of 
the product being extracted, produces the ordinate erected on that 
point where the diameter is cut. 

Sec. 214. As all the ordinates of the curve will be longer than 
the ordinate obtained, as in the last section, by the, distance from the 
given chord of the arc to the centre of the circle ; when it is sub- 
tracted from the ordinates extending from the centre of the circle 
to the curve, the remainder will be the length of the offset ordinates 
in the table. 

Sec. 215. " The angle at the centre of the circle, of which the 
rail-road curve is an arc, is double the angle of deflection of its chord 
line from the tangent." This principal has been so often referred 
to, and is so important to the engineer, that (contrary to my general 
plan,) I will here demonstrate it. 



Ill 

Make the said chord line the base of an isosceles triangle ; and 
let the radii, limiting it, be the two equal sides of the isosceles. 
And let the angle at the meeting of the radii in the centre (say 16°) 
be subtracted from 180° — leaving 164°. Halve this, making 82° 
for each of the base angles. Now it is manifest, that if 82° be sub- 
tracted from 90° (the angle formed by the tangent and the radius,) 
it will leave 8°, the angle between the tangent and the base of the 
isosceles triangle ; which is the assumed chord of the curve (or arc) 
proposed. 

Sec. 216. Cases may occur, where a cuitc may be required of 
the form of the long side of an oval. In such a case proceed in all 
respects as with a circle, until the radius and general chord line are 
found. Then shorten the radius by calculation, at the middle of the 
curve, as far as may be required. Consider the remainder of the 
radius as half the conjugate diameter of an ellipse. Also, the whole 
radius, doubled, as the transverse diameter. Offsets from the trans- 
verse diameter (ordinates) may be calculated thus : As the square 
of the transverse diameter, is to the square of the conjugate; so is 
the rectangle of the two abscisses of the transverse diameter (sup- 
posed to be cut where the offset stands,) to the ordinate or set-off. 
(See farther explanation of the ellipse hereafter.) 

Sec 217. Rail-road curves must not be too short, on account of 
the friction of flanges, and danger of running off ; and experience 
must limit the highest admissable degree of curviture. They are 
generally compared by length of radius. But in absolute strictness 
they ought to be compared by a method, in part analagous to the 
principle on which the movements of planets in their orbits are com- 
pared — ^that is, similar areas with proportional lengths of arc. The 
arcs of the areas may be thus compared : Having first found the 
radius, length of the arc, and area, of the tried railway, double that 
area, and divide the sum by the radius of the proposed rail- way ; 
which will give the length of the arc. The proportional lengths of 
the arcs wUl give the most simple and direct method of comparing 
them, with a view to their fitness, — ^the longest arc being propor- 
tionably the most curved and most objectionable. 

Sec. 218. The Convexity of the Earth is such, that the level- 
uig instrument, when pointing to a great distance, will cause a line 
'0 rise above the true level— theit is, the line will be 7.9 inches far- 



112 

ther from the centre of the earth at the distance of a mile, than is 
necessary to constitute a level. In truth, by a level we mean a 
curve, which if continued, will form a circle around the earth, every 
where equi-distant from its centre. At the distance of two miles, 
the levelled line will differ from the water level of the earth 2 feet 
7.9 inches — at four inlles distance, 10 feet 7.3 inches — at eight 
miles distance, 42 feet 6.6 inches.- These calculations were made 
on the supposition, that a tangent line nearly coincides with the 
150 thousandth part ofa circle (as 42 feet 6.6 inches amount to 
about that proportion of the earth's periphery.) Therefore this rule 
will be sufficient for all cases in practice. Square the semi-diameter 
of the earth, and the superficial measure of the distance run, sepa- 
rately — add these squares, and extract the root of the sum. This 
will give the length ofa line from the centre of the earth to the le- 
velled line (tangent.) Subtract the semi-diameter of the earth from 
that obtained line ; and the remainder will be the perpendicular 
height of the end of said line. 

Sec 219. If the length ofa degree of latitude (69.1 miles,) be 
calculated by the square root, according to the preceding rule, it 
will give 3211 feet 3.5 inches — whereas the true calculation by 
sines, &c., gives 3192 feet 9.7 inches. The error then in a degree 
of latitude will be about 18 J feet. 

Sec. 220. If perfect accuracy is required in great measured 
lengths on the earth's surface, as 10 degrees of latitude, find the 
true length of the tangent line, and its elevation above water level, 
thus : Turn the measured length into degrees, by saying ; as 25000 
miles gives 360°, what will 691 miles give ? Answer 10°. Now 
take this 10° for the angle at the centre of the earth, between the 
two radii, limiting the measured degree. This gives an isosceles 
triangle, with 10° at the apex and 85° at each of the other angles. 
Subtract the 85° from 90°, which gives the angle outside of the isos- 
celes, between its imaginary chord-line base and the tangent. Then 
subtract the other 85° from 180°, which gives the angle outside of 
the isosceles triangle, between said imaginary chord line, and the 
secant extending the radius line up to the tangent. As the chord 
line at the base of the isosceles triangle is found by the given pro- 
portions of it, it follows, that all the angles and one side of the out- 
side triangle being found, the true length of the tangent and radiu.g 



113 

extended may be found. Fi-om the extended radius subtract the 
true radius, which gives the elevated end of the tangent with accu- 
racy, 

MEASURING EXCAVATIONS AND EMBANKMENTS. 

Sec. 221. Under the head of mensuration, the inethod of calcu- 
lating a parallelepiped, a pyramid, the frustrum of a pyramid, and 
a triangular prism (including the wedge,) were shewn. Engineer 
D. C. Lapham, has shewn us [Sil. Jour. v. 27, p. 128,] how to ap- 
ply Professor Day's eighth problem in mensuration, so as to give a 
solution in cases, where part or all of these solids are found combin- 
ed in a prismoid embankment, or in a prismoid mass of earth to be 
excavated. In truth it is a rule of most extensive generality, apply, 
ing in all cases where there are straight sides ; or where sides can 
be equibly averaged so as to approximate plane faces. Wagon- 
loads of coals with boxes sprung between the stakes, heaps of rough 
stone, ledges of rocks required to be cut down, basaltic hills, trunks 
of rivers, &c., may be calculated by it, with more accuracy than 
by any other hitherto discovered rule. 

Sec. 222. Rule. Dimde the mas.? to be measured into so many 
sections, or prismoids, tliut each side shall le nearly straight from 
one end of each section, to the other. Find the area of the middle 
of each, and of both ends. Take the middle area four times, and 
each of the end areas once. Having added these six areas, multiply 
the sum by one sixth of the length of the section — this gives the solid 
contents. 

Sec 223. Without giving a full illustration, it will be a leading 
thought towards an illustration, to refer to the v/edge and frustrum 
of the pyramid. If the edge of the wedge is wider or narrower 
than the head, the width of the edge and of the two corners of the 
head must be added, to obtain an average of this modification of the 
triangular prism. Double the area of the triangular wedge is ob- 
tained by muUiplying the length by the thickness of the head. 
Thus, having double the area, and three times the length, (calling it 
a triangular prism,) by multiplying these dimensions together we 
obtain six times the soHd contents. Therefore it must be divided 
by six. The frustrum of a pyramid is a parallelepiped, four wedges 

15 



114 

and four pyramids. If the mass to be calculated is a parallelepi- 
ped, to take its area six times and then divide it by six, will produce 
the same result, as if but one area was to be used. But if the mass 
requires a wedge to be sliced off, to reduce it to a parallelepiped, the 
six areas will include the wedge, and not alter the calculation of 
the parallelopiped. As all masses with straight sides, &c,, (and 
which can be reduced to such by judicious averaging,) may thus 
be calculated ; this seems to be an exceedingly useful rule to the 
engineer. For it applies equally well to calculating the supply of 
water per second, &c,, by a running stream — considering the sur- 
face of the water equivalent to the level bottom of a canal ; and the 
iottom of the stream as equivalent to the uneven surface of a section 
of a canal. 

Sec. 224. In taking the transverse areas of the masses to be 
measured, the levelling instrument is essential. A plain is assum- 
ed as the bottom of a canal or as the basis of a rail-road, &c., to be 
excavated ; and its assumed depth is to be estimated from a fixed 
chair, (as it is technically called.) This consists of a stake driven 
strongly into the ground, or some other permanently secured object ; 
intended as an index of reference, of a known height above the level 
of the bottom plain of the canal, &c., to be excavated. Stakes are 
set in the centre of the canal or rail-road ground. 

Also in transverse sections, where areas are required to be cal- 
culated. These stakes have marks upon their levelled points, shew- 
ing their respective elevations and depressions, in relation to the ob- 
ject of their being set up. One hour's shewing, with the instru- 
ments in hand, is of more value than many days of reading. The 
manual use of instruments I shall not attempt to describe. A few 
general directions will be given in the proper place. 

Sec 225. In calculations for obtaining cross-areas, we rely upon 
these two propositions. 1st. Areas of trapezoids are found by add- 
ing opposite parallel sides, halving the sum, and multiplying the half 
sum by the distance between them. And, 2d, that a triangle made 
upon any line, wiil not change its area by moving its apex to any 
point on an opposite parallel line. [See sec. 33, 1 and 2.] If an 
excavation for a canal, diverging upwards, is to be made along a 
side-hill, so that the bottom will be but a foot or two below the sur- 
face of the earth at the lower side, and eight or ten feet \telow the 



115 

surface at the upper side, the transverse area may be found thus : 
1st. Cut ofFa trapezoid at the bottom, up to the level of the surface 
of the earth at the lower side ; and cast its area by the first propo- 
sition above. 2d. Find the level of the surface of the earth at the 
upper side ; and multiply the upper side of the trapezoid by half the 
distance between it and the line of the level, which will give the 
area of all above the trapezoid. 

Sec. 226. If the earth is undulating or ridgy, these rules may be 
applied so as to meet every case, with a little exercise of the inven- 
tive powers. For example : After taking off the trapezoid, as in 
the last section, if there is a ridge at the surface, take the level of 
its top and consider it the apex of a triangle whose base is the upper 
line of the trapezoid. Then if earth is left on one side of the apex, 
or even on both sides, the upper level may be considered as the base 
of a triangle, with its apex on the upper line of the trapezoid. And 
in some cases a trapezoid may extend to the level of the highest 
ridge, or knoll, and be cast as such. Then the vacant places be 
cast out or subtracted by making triangles or other regular figures, 
based upon the highest level. 

Sec. 227.* The measurement of excavation, embankment and 
masonry on rail-roads and canals in this State, is usually reduced 
to cubic yards ; while the latter item in some of the Middle States, is 
represented by perches of 25 cubic feet each. The method of de- 
termining on a side hill, the point at which the slope of excavation or 
embankment would meet the surface of the ground, was explained 
under the head of Staking out. At the same time, all notes neces- 
sary for the calculation of cubic yards in excavation and embank- 
ment, are taken and entered in the field book, as follows, viz. : 



Stat. 


Dist. 


Left. 


Centre. 


Right. 


Dist. 1 Cut. 


Cut. |Dist. 


40 
41 


100 
25 

75 


14.60 
18.40 
13.00 


+4.6 

+ 8.40 
+ 3.00 


+ 3.50 

+6.25 

+ 2.38 


+2.00 
+ 4.62 

+ 1.75 


12.00 
16.62 
11.75 



Sec 228. When the ground is level, the point at which the slope 

* The four succeeding sections were furnished by Engineer Evans. 



116 

will come to the surface, is found by merely adding the cutting of 
the centre to one half the width of the road, and laying off the dis- 
tance at I'ight angles from the centre, if in excavation ; but if in em- 
bankment, add 1^ the centre filling to one half the width of the road, 
and lay off as befoi*e. This is correct only when the ground is a 
plain. For if the surface slopes transversely of the line, it is plain 
that the filling (if filling it is,) will be greater on the one side and 
less on the other, as you depart from the centre line. And if the 
filling is greater, the width of the base, or the distance from the cen- 
tre to the outside of the embankment, will be increased in proper- 
tion to the slope of the bank — which we have before said, should be 
l^^ to 1 in ordinary cases. The object then of staking out, under 
the circumstances above described, is to ascertain the point at which 
the distance from the centre equals once and a half the filling at 
that point added to one half the proposed width of the road-bed. 

Sec. 229. It frequently happens on side hills, that there is filling 
at the centre stake, but within a few feet the ground rises so much 
as to require cutting. When this is the case, it is necessary, not 
only for calculation but also for the convenience of the contractor 
in commencing work, to ascertain the distance from the centre at 
which the cut commences. Suppose the instrument set as before 
described under Staking out, and the elevation by grade at each sta- 
tion also known. Subtract the elevation by grade of the station in 
question from the height of the instrument, and set the target to cor- 
respond with the difference. Let the target-man then hold his rod 
upon the ground at a short distance from the centre, and move up 
the hill at right angles to the line, until the hair of the instrument cuts 
the vane, and the place where the rod then stands will be the point 
where the cut commences. The distance from this to the centre 
stake must be measured and noted. 

Sec. 230. It has been a universal practice on public works, to 
require contractors to haul the earth a given distance from an ex- 
cavation, before they receive pay for the same as embankment. 
This distance, however, is not by any means uniform, but varies 
greatly on works conducted by different engineers. On all our 
Slate canals, it is fixed vX 100 feet; while on many of the rail-roads 
in this State, it is extended to 500 feet ; but we consider a mean 
between the two to be preferable, and have therefore established it 



117 

at 300 feet. After calculating the whole amount of excavation and 
embankment; whatever of the latter item comes within this distance 
must be deducted. 

CANALS. 

Sec. 231. Under rail-roads I have included most of the calcula- 
tions required for canals. These calculations I shall not repeat. 
Therefore, the student is to expect but little under this head which 
appertains to the mathematical arts, excepting what relates to items 
where water is an agent. This article will, therefore, be chiefly 
devoted to general descriptions ; excepting that it will close with 
the necessary calculations on supply of water and filling and empty- 
ing locks. (See sec. 221—226.) 

Sec. 232. Navigation. The general term for all transportation 
or conveyance by water, is navigation ; which is divided into natu- 
ral and artificial. But as natural navigation scarcely comes within 
the province of the engineer, I shall take no farther notice of this 
distinction. 

Sec. 233. Moving bodies on water. The particles of water move 
over each other without much friction or with very little adhesion. 
Therefore heavy bodies move on the surface of water with little 
resistance. One man has moved 100 tons 7 miles in one day. 

But as all heavy bodies sink into the water which sustains them, 
until they displace a measure of water of a weight equal to their 
own weight ; it is manifest that a volume of water forward of the 
moving body must be displaced by it. It follows that the form of 
the body, and the place to which the water is to be removed, are 
important subjects for the consideration of the engineer. Also, that 
a boat carrying 50 or 100 tons, will not add one ounce to the pres- 
sure on an aqueduct bridge, while crossing it. The law which 
governs ship-builders, in giving form to vessels, is manifest in com- 
paring the movements of a log or raft in water, with the sharp-built 
skiff or Indian canoe. 

Sec 234. Suppose the moving body, for example a crib-boat of 
lumber, to be 14 feet wide, and the canal the same width. Suppose 
the crib-boat sinks 3 feet into the water. The water before it is a 
wall 3 feet high ; all of which must pass back by the stern, when 



118 

the crib is in motion, through the thin crevices on each side and be- 
neath. The time required for this escape of the " wall of water" 
will be such as greatly to impede the progress of the crib. But 
were the canal 28 feet wide, the wall of water would escape late- 
rally where there was but little resistance. Still the banks would 
present some resistance ; as the waters nearest to the boat would be 
met by the waters stopped by the resistance of the banks. Hence 
it follows, that great breadth of water is favorable to the movement 
of heavy bodies on its surface. This principle is tested by our canal 
boats, when they pass by an artificial basin. 

Sec, 235. General law for constructing boats or other moving bo- 
dies. The inclined plane and wedge are known to be mechanical 
powers, which give an advantage, directly as the length exceeds the 
breadth. Call the wall of water the resisting force, and horse- 
power, wind, &c., the power. Then the power will be to the resis- 
tance, as the greatest breadth of the boat to the distance from the 
place of its greatest breadth to the extreme fore-end, where it cuts 
the water. Consequently the longer the boat is, forward of its 
greatest breadth, the less power will be required to move it. But 
there are numerous other circumstances to be taken into view in 
practice, which every ship-builder understands. Such as, that the in- 
clined plane principle applies to the ascent of the fore-part of the boat. 
For example, the scow, which retains its full breadth from end to 
end, has the advantages of the inclined plane in its ascending sled- 
like fore-end. Mathematicians have said that water would present 
a solid resistance when compressed with a velocity equalling 18 
miles per hour. But this applies when a broad plain is presented ; 
as if a plank- work should be fixed to the stem-piece of a steam- 
boat of equal area to a transverse section at the broadest part. 

Sec. 236. Choosing the ground for a canal. In choosing the 
ground for a canal, where there is an opportunity to make a choice, 
the engineer should recommend to the directors the following con- 
siderations : 1. Course of the prevailing winds. 2. Kind of soil 
through which excavations are to be made. 3. Its termination in 
regard to commercial advantages. The illustrious Clinton told me 
he would not recommend a canal in the present age "which should 
terminate with its last excavation." Canals should be constructed 
for connecting navigable waters, or for connecting a navigable wa- 



119 

ter with a great coal-bed, mine, or quarry. Prevailing winds have 
not been duly estimated. Side winds, though favorable to all sailing 
craft, are very unfavorable to canal navigation. They always drive 
boats ashore, and give no assistance to its progress. Winds in the 
direction of the canal present an opposing force and an accelerating 
force, which counterbalance each other on the whole. Northerly 
and southerly winds being most prevalent in America, east and west 
canals are not so favorably situated in this respect. It is well un- 
derstood by all boatmen, that winds are a less impediment (they are 
always an impediment) on the Champlain than on the Erie canal. 

Sec. 237. Excavations made in plastic clay, marly clay, marine 
sand and crag, are found to be permanent. In truth, all stratified 
detritus, called tertiary formation, make good beds and banks for 
canals. The marine sand (bagshot sand) would seem to be unfit 
for canal embankments ; but the trials at Irondequoit and Holley, 
on Erie canal, prove its fitness. 

Diluvial and post-diluvial detritus are too variable in character 
for any general rules, excepting that ultimate diluvian and anallu- 
vian are good materials for canal beds. 

Sec. 238. Agriculture and healilu Canals should be constructed 
with a view to health and agricultural operations. For if they per- 
mit water to ooze through their embankments, health and agricul- 
tural operations are injured. Therefore diluvian containing vege- 
table matter should never be used. 

Sec 239. Canal hanks, when they are not paved, should be 
bound by vegetables with creeping roots ; particularly when the 
canal runs through diluvian, as from Oriskany to near Genesee 
river. Agropyron repens (quack grass) is probably the best of all 
American plants for this purpose. It will prevent the production 
of unhealthy gases, and prevent the banks from shding down in the 
spring of the year. Banks are secured with well set paving stones 
where such stones are conveniently obtained. But the quack grass 
is best in wet places. 

Sec 240. Stop-dams were formerly made in canals each side of 
every place liable to failure ; so that when it gave way the naviga- 
tion would not be interrupted each side of the breach, nor the breach 
be enlarged by a long continuance of the flowing of water. These 
dams consisted of planks hinged to a bedded timber in the bottom of 



120 

the canal, so placed that a strong current would raise them up. Thus 
a breach in the canal by creating a current would stop itself. They 
are in some measure discontinued at the present time. 

Sec. 241. Aqueduct bridges are bridges supporting canals which 
are carried over vallies, I'ivers, &ev, the canals being, constructed 
of wood or stone. They are but half the width of the canals, as the 
boats are never to meet on them. 

Sec. 242. Culverts differ from aqueduct bridges in preserving the 
equal breadth of the canal, and in being constructed of earth, like the 
rest of the canal. 

Sec 243. Waste-weirs are openings on the sides of the canal, 
placed at a guaged height, so that the water will waste or flow out 
of the canal when it would otherwise be so high as to injure some 
part of the works. (See waste-weir under Water-Power.) 

Sec 244. Tow-path for men and teams to travel on when towing 
a vessel or raft which is floated in the canal. It should be three 
feet higher than the surface of the water of the canal ; and a tow- 
ing-rope should be 130 feet long. As teams must travel nights and 
days, and in rainy weather as well as in dry weather, the tow-path 
should be made of silicious or calcareous earth, that it may remain 
hard and even at all times. Its bed need not exceed three feet in 
width ; therefore the cost of a solid bed will be repaid in one season 
of boating. 

Sec 245. Cross bridges are made for changing sides with the 
team, at places where the situation of the canal has inducefl the en- 
gineer to change sides with the tow-path. These bridges are nar- 
row ; and the tow-path is so arranged, that the teams cross the 
bridges without having the tow ropes cast oflf. 

Sec 246. Heel-path. The side of the canal opposite to the tow- 
path, is called (by way of a pun upon the word tow — toe) heel-path. 
The engineer should make the heel-path as good as possible with- 
out incurring much expense ; for it is often almost as important to 
boatmen as the tow-path. Whenever a boat comes to for a night, 
or for a few hours by day, it must be on the heel-path side, and the 
boatmen will have frequent occasion to use it. 

Sec. 247. Location of locks is a subject of great importance. In 
ascending ledges, and in some other situations, the location of locks 
is fixed as a matter of necessity. But in most cases, the location 



121 

of a lock is a very important subject for the consideration of the 
experienced engineer. When the ground is so nearly level that the 
locks may be at any places within two or three miles, they should 
be separated as near the usual distances for changing teams as pos- 
sible. If locks could always be placed at the distance of six or se- 
ven miles from each other, this arrangement would be the most ad- 
visable. Long levels should never be sought. Every engineer 
concerned in laying out the Erie Canal, regrets having laid out the 
seventy mile and the sixty-two mile levels. By frequently agitat- 
ing the water of canals by passing it through paddle gates under 
great pressure, atmospheric air is united with it, and gives it health, 
ful briskness. The 83 locks of the Erie canal average four miles 
and a third from each other. The distance between many of them 
might have been proportioned better. For example, the nine locks 
at Cohoes Falls* should have been equally distributed to West 
Troy. This subject is at this day, (1838,) well understood ; and 
the change is now going on, by averaging the locks along the 
high ground at the right of the present locks. It was found that 
the basins between the locks were so much limited in extent, that 
boats were often grounded by drawing off the water for filling the 
locks. 

Sec. 248. Form of locks. The common form of locks is better 
than the elliptical form. And the size should never greatly exceed 
one boat in length and breadth, of the largest kinds which are to be 
used in the canal. For there is nothing gained by passing two boats 
at once, as the water required for filling is the same. Whereas 
there will be a great loss of water and time at large locks when but 
one boat is ready to pass. A still greater objection to large locks 
is, that larger and longer timbers are required for gates, which are 
heavy to move and more liable to be out of repair. 

Sec. 249. River locks. It is generally advisable to construct 
locks on the s'de of a river, out of the reach of freshets, when they 
are required for conveying a vessel around a fall or rapid. Thus 
the sloop lock in Troy (New- York) should have been located. This 
would have kept sloops out of the current from the fall of water 

* This article was publislied in 1830. 

16 



122 

over the dam ; and when above the dam they would have been out 
of the influence of the water-fall, or draft of water as it is called. 

Sec. 250. A lock consists of side walls, fall walls, hackings, wings, 
coping stones, recesses with holloio quoins, mitre sills, a pair of gates 
at each end, and each gate containing paddle-gates and a balance 
beam. Some locks have a crooked culvert in the wall for each pad- 
dle-gate ; and some have siphons in lieu of paddle gates. Two or 
four paddle-gates are most approved ; as one may be opened while 
the water is low before the gates, and the other after the water has 
risen to the paddles. Besides a paddle- gate way will weaken the 
main gate if large enough to let the water through expeditiously ; 
and it cannot be opened without great power, if large. 

Sec. 251. Laying out a canal. This may be conducted in all 
respects Hke laying out a rail -road, as directed under that head ; so 
far as respects all mathematical calculations and operations. In ge- 
neral there is not as much care required as to accuracy in turning 
curves, and minute levelhng by the inch or foot. But the line must 
be straight to a considerable distance each side of every cross 
bridge, aqueduct, lock, or whatever lessens the breadth of the ca- 
nal, by which boats would be liable to strike. 

Sec. 252. Reservoirs. If there is great necessity for depending 
on water for the summit level which will be deficient in the dryest 
seasons, the line should be so run as to afford a convenient place 
for a reservoir of sufficient capacity. The reservoir should be a 
very little higher than the canal level ; so that all the water not es- 
sential for practical use, may be saved in the reservoir to let into 
the canal from time to time as occasion may require. 

Sec 253. Water for supplying feeders. In laying out a canal 
the most important consideration is a sufficient quantity of water for 
feeding it. To ascertain whether or not the supply will be sufficient, 
observe ths following directions : How many boats will probably 
pass every day 1 For a column of water whose base is equal to 
the square of the area of the lock, and whose height is equal to the 
difference between the highest and lowest surfaces in the lock, will 
be required for every boat that passes to the lower level. But we 
may suppose that only one half of the ascending boats will require 
the same quantity. Because if a boat ascends first after one has 
descended, no water is to be taken into account. 



123 

Sec. 254. Allowance for filtrations and evaporations should be 
equal to about one-twentieth part of all the water used for feeding 
the canal. 

In addition to all the allowances made for the first lock on each 
side of the summit level, allowance must be made for each succeed- 
ing lock, which is supplied from the summit level. Perhaps one 
twentieth for each lock. Two methods are in use for supplying the 
succeeding locks: 1st. To construct a sluice-way by waste-weir, 
to carry the excess of water around the upper locks to supply the 
waste to the lower. 2d. To make every succeeding lock to fall 
about six inches short of the preceding, in depth. 

Sec. 255. Supply of water ly feeders. The quantity of water 
which passes any point in a flowing stream per second, must be as- 
certained before a decision is made in regard to its sufficiency as a 
feeder for a canal. A calculation must be made of the supply of 
water which the stream will afford when it is at its lowest, in summer 
droughts. The best method of calculation is, to consider the sur- 
face level of an assumed trunk of the stream, as equivalent to the 
bottom level of a canal excavation. (See sec. 221.) Then take 
transverse depths of the stream, and consider them as the upper le- 
vels for calculating transverse areas of excavations, as taken by the 
level. (See sec. 222.) Then proceed by the four middle areas, 
and the two end areas, in all respects as directed in excavations and 
embankments. (Read sections 221, 222, and 225, attentively.) 

Sec. 236. The cubical contents of the trunk of water being found 
as if it was a permanent solid, the time it occupies in passing over 
its lower limit must be ascertained. If the stream does not exceed 
about three feet in depth, branching limbs with plenty of leaves may 
be thrown into the upper end of the trunk, and repeated several 
times, for determining the velocity. The branches must occupy 
nearly the whole depth, as water flows faster near the surface. 
They must be thrown in so as to average the velocity from near 
the shores to the centre. In deep rivers an empty and filled bottle 
may be tied togethcB, so that one may float and the other sink. 

The velocity may be measured by a watch ; but a second pen- 
dulum 39.1 inches in length, or a half-second pendulum 9| inches 
is preferable; or 9.77, more accurate. 



124 

Sec. 257. Having obtained the result of the calculations of the 
two last sections, proceed to compare the quantity of the supply, 
with the quantity required for filling the locks the maximum of 
times required ; allowing the full measure for each filling, and also 
allowing for all the wastes described in sec. 254. 

Sec. 258. The time of filling and emptying locks is not connected 
with the general supply, any farther than to determine whether it 
will be too long for a reasonable delay of boats. But the paddle- 
gates for the admission and discharge, is a necessary subject of cal- 
culation. On referring to sections 139 and 140, and by an applica- 
tion of common sense, this calculation will be evident. But the dis- 
charge may require more particular calculation ; therefore a full 
description of a familiar example will be found in next section. 

Sec 259. Calculation of the time of emptying the east lock of the 
pair of locks at the junction of the Erie and Champlain canals. 





Ft. 


Length of the lock within. 


89 


Average breadth within, 


15.5 


Depth, 


14 


15.5 217 




14 89 




620 1953 




155 1736 





217.0 19313 cubic feet of water contained 

in the lock. 

Four paddle gates in the lower gate, each two feet square ; mak- 
ing each an aperture of four square feet. 

Reduce each aperture one third, on account of friction (adhesion) 
and contraction of vein. 

3)4.00 
1.33 

2.67 Reduced aperture. 



125 

Head is 14 feet ; but falling water diminishes the force of the head 
one half. Therefore the head is to be computed as if 7 feet high. 

7.0000(2.65 2.65 Square root of the reduced head. 
4 8 Velocity of one foot head. 



46)300 21.20 Length of effused trunk per second, 

276 2.67 Reduced aperture. 



425)2400 14840 

2225 12720 

4240 



56.6040 Water effused each second from each 
4 paddle-gate. 



226.4160 Effusion per second from all the gates. 



226.4)19313.0(84 Seconds, true answer. 
18112 



12010 
.10056 



Sec 260. The calculation made in the last section, was from 
measures and trials actually made in January 1838, by the engineer 
class of Rensselaer Institute. By starting all the four paddle-gates 
at once, and measuring time by the oscillations of a pendulum 39.1 
inches in length, they emptied the lock several times, in precisely 
84 seconds each time. By often repeating similar measurements 
and calculations, a very important branch of engineering will be- 
come familiar. Flumes of flouring mills, factories, &c., are calculat- 
ed in this manner ; excepting that the square root of the whole head 
is taken, as in all cases where the water is kept at a uniform head. 
(See sections 139 and 140.) 

ROADS IN GENERAL. 

Sec 261. Kinds of Roads. The principal distinctions among 
the roads of this country are, 1. Uniedded, most common roads, 
which receive their forms from the feet of horses and wheels of 
carriages ; or remain as they were when first laid out. 2. Turn- 



126 ■ 

fiked, when made of earth in the form of beds, descending into late- 
ral ditches. 3. McAdamized, when made in the same form of turn- 
piked roads ; but the materials consist wholly of pounded stone, in- 
cluded between curbs. 

Sec. 262. Laying out roads is too often under the influence of 
private interest; otherwise they might always be well laid in new- 
ly settled countries. But it is very difficult to change the location 
of a road in an old inhabited town. The ground chosen for com- 
mon roads, to be supported by tax, should be free from sloughs, and 
should avoid clay-beds as much as possible. This should be done 
at the expense of distance ; for the taxes are generally too low for 
raising large sums for bedding and repairing roads through such 
places. 

Sec. 263. Causeys and bridges. So many treatises are before 
the public on these subjects, that little remains to be said. They 
should never be at the foot of a steep hill, when it can be avoided 
by turning the road, or by digging down the hill. And causeys 
should not be made with large logs or large stones. For the earthy 
covering will soon be worked down among such coarse materials ; 
leaving them a naked nuisance. If large stones or large logs are 
laid in, and covered with small stones or saplins, the evil will thus 
be remedied. 

Sec 264. Breadth of causeys and bridges. On the score of 
economy, as well as of convenience, these should be wider than the 
custom is, in this country. Bridges will stand much longer for 
having wide abutments ; and causeys (especially if made of wood) 
will be more firmly fixed, if wide. 

Sec 265. Mile and guide-boards. These should never be made 
of stone. Good sound plank are better. Every one has observed 
the destruction of mile-stones by mischievous villains. Plank can- 
not be destroyed, nor even injured, without considerable labor. 
But a mile-stone is broken and destroyed by one stroke of an ax, 
or one stroke with a large stone. Guide-boards are always made 
of wood ; but a mere slip of a thin board, nailed to a post, is soon 
demolished. A single plank when but two roads meet ; or a trian- 
gle or quadrangle of planks when more than two, should be used, 
half-charred at the ends set in the ground. 



127 

Sec. 266. Planting, or leaving, trees. Trees should be left in 
laying out roads through woods, and should be planted out in all 
other cases, in such situations that the road may be shaded from 9 
A. M. to 4 P. M. from 1st May to 1st September. The best indi- 
genous trees m America are the red maple and sugar maple. Trees 
^ are of more value than is generally supposed. Take the following 
calculation : 200 trees may be set at 25 cents each — that is, the 200 
trees per mile will cost S50. The cost will be 82000 for 40 miles, 
a day's travel for a team. On all great thorough-fare roads, 40 
loaded teams will pass each day, for 150 days of hot weather. This 
gives 6000 day -journeys ; which would be but 33 cents each, if the 
whole expense must be paid the first year. If the public fund only 
is considered, we may safely say, that less than half a cent is paid 
each trip for the benefit of dense shades for 40 miles, in the oppres- 
sive heat of summer. For all such trees will endure 60 to 80 
years. 

Sec. 267. Watering places. The value of watering troughs far 
exceeds that of shades. Walking horses, which draw loads, should 
drink once in three miles. Watering places must be supported by 
wells, excepting in places where there happens to be a stream. To 
neglect watering places ought to be made a crime by statute, for 
which road commissioners should be indictable. 

Sec 268. Level roads. For loaded carriages when the horses 
walk, a level road is best. But for trotting teams a road is best, 
when moderately undulating. Even a hilly road, in such cases, is 
better than a level one. For a load pressing alternately on breast 
and breech, is easier for the horse. 

Sec. 269. Resting places. Loaded teams are greatly reheved 
by resting places on hilly roads. And even pleasure-carriage tra- 
velling is often benefitted by them. They are transverse mounds 
of earth, made smooth and sloping ; a little oblique to the direction 
of the road, that they may serve to direct the water of sudden show- 
ers into one of the ditches. 

Sec 270. Zigzag roads. Such roads are required on the face of 
steep mountains. Such a road takes seven tacks while ascending 
Catskill mountain to the lakes and ^Mountain House. The principal 
subject to be considered in la^-ing out such roads is, to give broad 
spaces for the turns. I laid out that road, under the direction of the 



128 

commissioners. At first without any regard to the breadth of the 
turning ground. But I was compelled to alter the whole arrange- 
ment, after the work was commenced, in the year 1807. 

Sec. 271. Dug-ways. When roads are cut into side-hills, like 
shelves, they are called dug-ways. One rule is never to be over- 
looked in such cases. It is, that the outer side of the road, from the 
hill, must always be about one foot higher than the inside. And it 
must not be forgotten, that all springs issuing from the hill, must be 
carried across, under the road, in very large sewers. This is ne- 
cessary to prevent an accumulation of ice-ledges across the road, 
which make it totally impassable. 

Sec. 272. Agriculture and health. The same rules apply to 
roads, which are applied to canals in section 238. It may be added 
that roads are so numerous, as to furnish the means of doing much 
good in this respect. All the individuals in society make up the 
whole of community. Therefore if the property of one is benefitted 
the whole body is benefitted. Numerous cases occur to an honest 
board of road-commissioners, for doing much good. All conductors 
of water may be directed so as to benefit the nearest farmer, with- 
out any injury to the public. So in passing fai'm houses, there are 
numerous ways for an accommodation. Stagnant waters should 
never be allowed to settle down from a road near a dwelHng house. 

Sec. 273. Hills and mountains. The ascents and descents of 
hills, are important subjects for road-commissioners. By attending 
to the laws of the inclined plane, and to the balancing principle appli- 
ed by the horse in drawing a load, common sense will suggest rules of 
practice. When a horse is drawing to the extent of his strength, 
his hind feet form a pivot upon which the weight of his body is ba- 
lanced against the resistance of the load. Should the hill be so 
steep, that the centre of the gravity of the body of the horse, is di- 
rectly above his hind feet, he can draw nothing. Reduce the steep- 
ness of the hill, and the weight of the horse will apply in an increas- 
ing ratio. This ratio will have the advantage of the lever also. 
The ascent of 18 inches to the rod is the limit imposed by the legis- 
lature of the State of New-York, on several turnpike companies in 
mountainous districts — that is, an ascent of 5 degrees. If the 
ascent exceeds 6 degrees, it is a tiresome road ; and commissioners 



129 

ought to avoid any farther increase in the ascent, by the zigzag 
form. (See sec. 270.) 

Sec. 274. Angle of friction in the movement of carriages on dif. 
ferent roads. The angle of friction is estimated by placing a car- 
riage on an inclined plane in the road to be tested, where the de- 
scent is just sufficient to give the smallest degree of motion. The 
motion must not be sufficient to acquire any increased velocity by 
its progress. A section of this inclined plane is to be considered aS 
radius and the height of the elevated end as the sine of the angle of 
ascent. One ton for a load on a M'Adam road, will move with the 
radius 50 to a sine of one by measure — on a very smooth pavement 
the radius of 68 to a sine of one. On a level road of the same qua- 
lity, the same proportional of weight, suspended over a pully, as the 
sine to the radius, will start the load. Thus about 40 pounds sus- 
pended over a pully will start a ton load on a level, if the carriage 
is of the best construction in regard to friction. 

Sec. 275. Location of bridges. Two considerations must always 
govern in the location of bridges, if any choice can be exercised. 
1. It should be located below a natural ice-break if possible ; as a 
fall of water, &c. 2. It should be placed where the abutments may 
be secure ; for where abutments can spread, the bridge, if arched, 
is never secure. 

Sec 276. String. pieces. When a bridge is made by planking 
upon straight string-pieces, they must be strongest in the middle. 
To be largest is not always to be strongest. If the grains of the 
timber are straight, and the top of the string-piece straight on the 
top, it will be but little stronger for swelling underside so as to be 
much thicker in the middle. For its increase in strength depends 
on the lateral adhesion of the fibres, which is feeble in the straight- 
est and most thrifty growing timber. But if the timbers swell out 
at their sides, this objection will not apply in so great a degree. 

Sec. 277. When string-pieces are supported on bents or piers, 
some calculation is required in setting off the distances between the 
supports, or in selecting timbers of the most suitable length. By 
referring the mind to stations taken by men, when carrying long 
and heavy timber, common sense will have a sufficient guide in this 
matter. For example : 6 men are to carry a heavy beam, which 
is 60 feet long. To place these men in pairs, so that each pair may 

17 



130 

support equal portions of the weight, their carrying-sticks must be 
so placed that each pair shall bear 20 feet. If the timber was cut 
into three 20 feet pieces, and each stick was put under the centre of 
each 20 feet, it is manifest, that the weight of the whole 60 feet 
would be equally distributed. To arrange the three pieces in one 
line, touching end to end, would not alter this proportion — ^neither 
would pinning, or otherwise uniting them. Therefore the first pair 
would be placed 10 feet from the fore end — the second 30 feet — the 
third 50 feet. If two of the men were placed at the fore end, and 
the other four were to lift at one stick, that must be placed 45 feet 
back. This would leave 15 feet back to balance the 15 feet be- 
tween it and the centre. The pair at the fore end, would, upon the 
lever principle, lift but half the other 30 feet, if the back half was 
cut off and the other stick placed at the new cut end. But after all 
the back half is neutralised by the balance of the part back of the 
hind carrying-stick, the two sticks are applied to the fore half with 
different levers by 15 feet. That is, the centre of the weight of 
what is not balanced, is 15 feet from the forward carrying-stick, and 
30 feet from the back carrying-stick. Therefore the forward stick 
will support two thirds (20 feet) and back stick one third (10 feet) 
which, added to the back 30 feet, gives the true proportions. 

I preferred this method of illustrating the whole doctrine of beam 
pressure to that of givmg a set of rules. Not only bridges, but 
flumes, mill-dams, locks, house-beams, &c., &c., require an atten- 
tion to this subject. 

Sec. 278. Repairing roads. It is astonishing that our highway 
masters and turnpike companies, still continue to fill up ruts with 
loose soil ; when they see the first wheel that follows such repairs 
restore the ruts. It is still more absurd to fill ruts with stones, wood, 
&c. But one efficient method has hitherto been adopted. It is to 
fill the ruts with the same material of which the road consists, by 
pounding it down in a succession of thin layers, until it is consider- 
ably harder than the rest of the road. This makes a durable repair 
and the labor is not great. Two men with sledge-hammers, one on 
each side, will pound the fourth of a mile of bad ruts in one day ; 
while one man with a cart and one with a spade will supply the 
filling. 



131 



WATERWORKS. 

Sec. 279. The term Waterworks is applied to the conduction of 
water through pipes or raceways, (mostly through pipes,) where 
water is to be used as an element ; not for its mechanical force. 
The principles of waterworks, as a science, are not generally stu- 
died, and, of course, are little understood. 

Sec. 280. We have but one American treatise ; and that has, 
probably, no equal on either continent, as a concise digest of all that 
- is valuable on the subject. I mean E. S. Storroio^s Treatise on 
Waterworks. To those mathematicians who wish to go deeply into 
the subject, and to trace this truly experimental science back to its 
origin, then to follow down its history to the present day, this little 
duodecimo of 242 pages, is most emphatically recommended. He 
does justice to the early investigations (in 1771) of Abbe Bossus — 
of Chezy (1775)— of Dubuat (1786)— of Coulomb (1800)— of 
M. de Prony (1804.) But the German, Eytelwcin, may be said to 
have given the last finish to the formulae now in use, in 1814 and 
1815, through the Memoirs of the Academy of Berlin. 

Sec. 281. Waterworks being a subject not commonly studied by 
American engineers, I shall here give a few essential formulae, 
with a very general statement of the theory. 

Sec 282. Water moves in pipes or raceways, under the govern- 
ment of ACCELERATING FORCES and RETARDING FORCES. In Strict- 

ness, there is but one accelerating force ; which is the head waters 
above the place of discharge. And there is but one retarding 
force ; which is the adhesion against the sides of the conducting 
PIPE, or raceway. But in calculating practical results, it appears 
to be necessary to take into view the area of the pipe, or raceway, 
the interior surface of adhesion, and the length of the raceway, or 
pipe. 

Sec 283. It has already been shewn [see sec. 139 and 140] that 
the force given by the head waters, is as the square root of the 
height. But the retarding power of adhesion (sometimes called fric- 
tion) depends, for the estimate of its influence, solely on trial. As 
different calibres of pipes, and different measures of sides and bot- 
toms of raceways, present different proportionals of surface for ad- 



132 

hesion, nothing but very extensive series of experiments could fur- 
nish rules, or formulae, for practical use. 

Sec. 284. Though the fall of water (that is, the elevation of the 
head above the place of discharge,) is elementarily the only accele- 
rating force, and adhesion to inner surface of the conducting pipe or 
channel, is the only retarding force ; yet it is found that several mo- 
difications and combinations of these elementary principles, must be 
taken into all calculations — for the reasons read Storrow. 

Sec. 285. In pipes of Avood, iron, earthen, or whatever close con- 
ductors may be used, the following combinations and modifications 
are necessary : 

1. Diameter of the pipe. 

2. Difference of level between the head and discharge of the 
water. 

3. Length of the pipe — consequently the continuance of adhe- 
sion. 

4. Difference of level divided by the length of the pipe. 

5. Velocity per second. Here each section is considered as the 
acquired velocity, or the successive retarding results ; and may be 
estimated in infinitely small divisions. 

6. The quantity discharged per second in cubic feet. 

The area of a section of the pipe, as divided by the inner peri- 
meter, requires some calculations. This quotient is called the viean 
radius. It may be perceived, that the larger the area the greater 
will be the velocity, but the greater the perimeter the less the velo- 
city. In truth, after the perimeter is increased to a certain propor- 
tion, compared with the area, it totally overcomes the area ; and 
the water stops flowing, by the principle called capillary attraction. 

Sec. 286. Formula for fipes when quantity discharged is sought. 
The engineer is called on to answer this enquiry : How much water 
will be discharged per second, if the head (above the place of dis- 
charge) is 70 feet, the diameter of the pipe 10 inches, and the 
length of the pipe 900 feet? 

Prepare for the rule by reducing the feet to inches. 

Rule. 1st. Multiply the diameter by 57, and that product by 
the head (70 feet) and set this last product down for a dividend. 
2d. Take the first product (diameter multiplied by 57) and add to 



133 

it the length of the pipe (900 feet) and set this sum down for a divi- 
sor. 3d. After the division, extract the square root of the quotient. 
4th. Multiply that root by 23.33— the product will be the velo- 
city in inches per second. 5th. Find the area of the pipe at the 
place of discharge, and multiply that by the velocity in inches per 
second. This product is the quantity of water discharged per 
second in cubic inches. 

Sec. 287.* Formula for pipes lohen the diameter is sougJit. What 
diameter of pipe will be required to discharge 3 cubic feet of water 
per second, if the head is 10 feet and the length of the pipe is 4000 
feet? 

Prepare for the rule by reducing all the measures to feet and 
decimals of feet. 

Rule. 1st. Square the cubic feet (3) multiply the square by the 
length of the pipe (4000 feet) and take this product for a dividend. 
2d. Multiply the head by the square of 38.116, (10 x 38.116 x 
38.116=14528.29) and take this product for a divisor. 3d. After 
the operation of division, extract the root of the quotient to the fifth 
power, by sec. 30 ; which will give the diameter of the pipe in feet 
and decimals of feet (1.199.) 

Sec. 288. Formula for open canals when the velocity and quantity 
are required. How much water will be delivered per second, if the 
area is 4.8 feet (2.4x2) head 10 feet, perimeter inside 6.8, length 
30 feet? 

Prepare for the rule by reducing all the measures to feet and 
decimals of feet. 

Rule. 1st. Multiply the area and head together for a dividend. 
2d. Multiply the perimeter and length together for a divisor. 3d. 
After the operation of division, multiply the quotient by 9582, and 
to this add 0.0111. 4th. Extract the square root of the last sum. 
5th. To the root add 0.109. This will give the velocity of feet per 
second. Multiply the said feet by the transverse area of the trunk 
of flowing water, which will give the quantity in cubic feet. 

Sec. 289. As open canals present their flowing waters to the eye, 
their laws of motion are subject to more convenient inspection than 
those of pipes. At the Rensselaer Institute the students in civil en- 

* This, and three other examples, were obUgingly calculaled for my pupils by my learned 
friend, Mr. Slorrow. 



134 

gineering have generally obtained the following proportional results, 
or nearly so. A raceway, smoothly planed within, with exact slid- 
ing gates 36 feet apart, is used. It is 2 inches wide and 2^ deep 
within. When the raceway is so inclined that the upper gate is 56 
inches higher than the lower one, the water flows from gate to gate 
(36 feet) in 6 seconds. If the upper gate is drawn 2 inches, the 
velocity so far contracts the flowing trunk of water, that the lower 
gate precisely touches the surface when it is drawn 1.1 inch. There- 
fore the increased velocity, under an angle of 7° 23' inclination, 
diminished the volume of water nine twentieths in flowing 36 feet. 

Sec. 290. These experiments may not be perfectly accurate. 
But they approximate truth near enough for general illustration ; 
and students of all schools should repeat them. The law of falling 
bodies, as illustrated in sec. 140, and the law of the inclined plane, 
as in sec. 152, should be referred to in explanation of this experi- 
ment. Students will not overlook the difference between pipes and 
open raceways, caused by the water on the upper side, in the latter 
case, not being subject to adhesion. Of course the perimeter in- 
cludes the bottom and two sides only ; whereas pipes present ad- 
hering surfaces on all sides, with additional resistance from adhesion 
on account of increased pressure against the whole inner surface. 

Sec. 291. As almosplieric pressure often has more or less influ- 
ence upon the flowing of water in pipes and open raceways, the stu- 
dent is referred to sections 141 — 144, where the most important 
principles are explained. The pressure averaging about a ton 
weight to a square foot near tide-water level, it often becomes a sub- 
ject deserving particular attention. But on high mountains the 
pressure of the atmosphere is greatly diminished. It even becomes 
so rare at the height of about 45 miles, that it does not reflect the 
sun's rays sufficiently to become visible in the state of twilight. 

Sec 292. The highest point where the atmosphere is sufficiently 
dense to reflect light, may be found as follows : 

1st. Take the time, by a good watch, between the disappearance 
of the sun in the western horizon, and the disappearance of twilighto 

2d. Calculate this ti7ne, in the manner hereafter explained, so as 
to ascertain what the time would be (or is) where the sun goes down 
vertically (as at the equator on the 22d of March.) 



135 

3. Then say, as the minutes of 24 hours to 360 degrees, so are 
the minutes, vertically taken, between sun-set and twiUght-set (about 
70) to an angle at the centre of the earth, formed by a radius to 
said centre from the observer, and from the place on the earth 
where the sun disappears, where the observer sees the last depart- 
ing ray of twilight in his horizon. 

4th. Take half said angle at the centre of the earth, for one of 
the acute angles of a right angled triangle, formed of the observer's 
horizon, his vertical semi-diameter of the earth, and a line from the 
centre of the earth to the point of the last appearance of twilight. 

5th. Then say, as the co-sine of the half angle at the centre of 
the earth, is to the semi-diameter of the earth (about 4000 miles,) so 
is radius (the angle at the observer) to the distance from the centre 
of the earth to the point of the last appearance of twilight. 

6th. Subtract from the last answer the semi-diameter of the earth 
(4000 miles) and the remainder will be the height of the atmos- 
phere, where it is just dense enough to reflect and refract the sun's 
rays sufficiently for rendering it visible to the earth's inhabitants. 

Sec. 293. It may be desirable to the correct student, to understand 
a practical method for determining the true time to be assumed for 
the calculations of the last section, between the disappearance of the 
sun's face, and the disappearance of its last departing rays of twi- 
light. The plainest practical method, and the one best adapted to 
student's practice, is as follows : 

1st. Set a compass to find the bearing of the point of the sun's 
setting, and note the time of its setting. 

2d. In the same manner, note the time and bearing of the obtuse 
apex of the last departing rays of twilight. Thus you have the dif- 
ference of time between the disappearance of the sun's face and of 
twilight. 

3d. Call the difference of time between the setting of the sun and 
the setting of twilight, the hypothenuse of a right angled triangle. 
Call the horizontal angle between the point where the sun sets and 
where the day-light sets, the horizontal leg. For the vertical leg, 
(the answer required,) apply the rules for right angled triangles. 
Also remember to turn time into angles, as in all cases where 24 
hours give 360 degrees, &c. 



136 

Sec. 294. The said vertical leg may be found without taking the 
bearing of the points at setting of the sun and twilight, by finding the 
angle which the sun makes with the horizon at setting. This may 
be done by calculation, made from the latitude of the place of obser- 
vation and the sun's decHnation. 

Sec. 295. As aqueous vapor diminishes the specific gravity of the 
atmosphere, it often becomes a subject of consideration — particularly 
in the use of the barometer, and in calculating for the ascent of 
water in pipes in passing over hills, &c. Vapor being visible in 
the form of clouds or fogs, it may be well for the student to give 
his attention to the natural history and heights of clouds, for part of 
each day during one week ; as clouds are lighter than air. 

Sec. 296. Five forms of clouds often precede each other in regular 
series. In fair weather during summer months, the stratose clouds, 
usually called fogs, often appear in the morning near the earth. 
After the sun shines upon them, they ascend in a state scarcely 
visible, and at length form the cumulose clouds. These are the 
bright shining clouds in brilliant heaps above, with apparently 
straight bases below, when viewed horizontally. They ascend still 
higher later in the day, and form the cirrose clouds. These have 
a fibrous flax-like appearance, and rise the highest of all clouds. 
At length they descend more or less, and become the cirro -cumulose 
clouds, by assuming a knotted or curdled appearance at first, and 
then becoming confluent. They either become stationary, produc- 
ing rain or snow; or break up, and their fragments become 
cirro-stratose clouds. These are the patches which have a stratified 
appearance when viewed horizontally ; but they never approach 
the earth, like fog. No rain falls from this series of clouds, except- 
ing while in the cirro -cumulose form. 

Sec. 297. Three forms of clouds seem to he independent of all 
other forms, and of each other. The nimhose cloud, generally call- 
ed the thunder cloud, as soon as it commences forming, begins to 
move pretty uniformly and steadily. At first it exhibits a heaped 
top, like the cumulose cloud ; but as it advances in size, it shoots 
forth a kind of spray-like form from its uppermost heads. It usu- 
ally produces rain, and breaks up soon after. The villose is a kind 
of open fleecy cloud, called scud, which moves with great rapidity, 
often in a direction different from the clouds above. It is generally 



137 

formed suddenly, and breaks up suddenly. The only remaining 
variety is the cumuh-stratose clojud. It is very rarely formed, and 
always appears to rise up in the horizon like the smoke from a fur- 
nace. Its top generally seems to pass into a cirro-stratose cloud 
above, and there spreads out like the top of a mushroom ; it is there- 
fore generally called the mushroom cloud. 

All snow storms and settled rains proceed from the cirro-cumu- 
lose, and all hail storms and showers, from the nimbose, clouds. 

Sec. 298. The height of a nwibose cloud may be taken as shewn 
in the following example : May 30th, 1837, during a severe thunder 
shower, I suspended a pendulum near the west door of the Institute, 
and directed an assistant to watch its vibrations, while I observed 
the origin of three successive chains of lightning. The assistant 
noted, by the pendulum, the seconds between the flashes and the 
sound. The time averaged 21^ seconds — and the angle above my 
horizontal level was found, by the sextant, to be 11^°. Allowing 
1124 feet per second for the sound, the hypothenuse from that point 
in the cloud was 24166 feet. This gave the height of the cloud 
4818 feet above my level. The earth's convexity (after finding 
the horizontal leg) gave 13.3 feet. (See sec. 218.) Therefore the 
height of the cloud was 4831.3. But my level was 73 feet above 
the tide-water of the Hudson — of course the cloud was 4904.3 feet 
above tide- water level. 



WATER-POWER, 

APPLIED TO DRIVING MACHINERY. 

Sec 299. The elementary laws of water-power are explained 
and illustrated in sections 136 — 140. The student must attentively 
review those five sections, when he is about to be exercised in the 
present application of this power. I shall go no farther into the 
subject, than is necessary for preparing the student for those duties 
which strictly belong to the out-of-doors engineer. I mean, that he 
must be qualified to take the original measurements, and calculate 
the power of any proposed mill-seat, before the commencement of 
any of the works. In doing this, he takes flouring mills as his 
standard ; estimating their powers by the quantity to be floured in a 

18 



138 

given time. Then it is the business of the miU-wright and machinist* 
to make comparisons, and construct the works according to circum- 
stances. 

Sec. 300. The first step to be taken is, to take the necessary 
measures, and to calculate in cubic feet, pounds, or tons, the water 
which flows by any point in the stream per second. Directions for 
this operation are given under the head of Measurements of Exca- 
vations and Embankments, sec. 221 and 222. But it may be well 
to give more particular directions here. 

Sec. 301. To find the supply of water, select a time for taking 
measure when the stream is at its lowest, highest, or middle state, 
according to the object of the owner. Sometimes a mere flood-mill 
is desired — in other cases it is not desired as a drought-mill, &c. 
Select a trunk of the stream which is the most uniform in width, 
depth, and velocity, and traverse one shore with the compass. Its 
length ought to be such, that sticks, leaves, &c., will require at 
least 10 seconds to flow through it — 20 seconds will be better, if 
such a trunk can be found, that is nearly uniform in width, depth 
and velocity. The length being taken in feet proceed as follows. 

Sec. 302. Take measures for a transverse area at every mate- 
rial variation in depth, width, or direction, in this manner : Measure 
the breadth, and also the depth, at every material difference in 
depth — be particular to notice the distances between the places where 
depths are taken. Also measure the distances between the mea- 
sured areas — all in feet and decimals of feet. 

Sec 303. Take the velocity of the stream in this manner : Sus- 
pend a pendulum for beating seconds 39.1 inches in length — or use 
a watch with a second hand. The pendulum is preferable. Let a 
careful assistant note the seconds. If the water is shallow, throw 
in branching weeds, green bushes, &c., of such forms that they will 
be driven along by the action of the stream from the top to near the 
bottom."!" Note the seconds they occupy in running through, about 
eight or ten times, in the strength of the current. Then try the 
same experiment as many times towards each shore ; and use your 
judgment in estimating the proportional part of water in the trunk 

* Oliver Evans, and his editors since his decease, have prepared a work, under the title of 
Mill- Wright's Guide, which surpasses all commendation. It is a remarkable specimen oftlie 
union of science and art. 

t Shavings of white wax arc best. 



139 

where the side experiments were tried — always bearing in mind 
that the diminution of velocity as well as of the depth, are to be taken 
into view ; for though the diminution of depth will come into the 
calculation of areas, its retardation by adhesion at the shallow bot- 
tom and shore, must be separately estimated, as near as may be. 

Note. These being all the measures to be taken in the field, you 
will return and make the calculations. 

Sec. 304. A plot of the following kind, will greatly facilitate the 
operation. Fix on the scale by which your plot shall be made ; 
then draw two horizontal lines at a distance fx'om each other equal 
to the length of the trunk. Plot the traversed shore of the trunk, 
which will terminate in the parallel lines. Then lay off all the trans- 
verse sections, parallel to the horizontal lines, and the true distance 
from each other. Consider these lines as those drawn across the 
straight surface of the stream. Let fall perpendiculars from each, 
according to depth of the measures taken. Coimect the lower ends 
of these measures, which will exhibit each transverse area. These 
areas may then be calculated by Lapham's method, as described in 
sec. 225, or they may be cut up into triangles and trapezoids, as 
described under land surveying, sec. 94 and 95. 

Sec. 305. The cubic contents of each section of the trunk may 
then be cast, by adding the areas of the ends to four times the area 
of the middle, and multiplying that sum by one sixth of the length. 
(See sec. 222.) 

Sec. 306. Having found the cubic contents of the trunk in cubic 
feet, average the number of seconds, which the branches, &c., oc- 
cupy in passing through the length of the trunk. Divide the con- 
tents by the seconds, and the quotient will be the cubic feet which 
pass by the lower point in the trunk per second. Cubic feet may 
be reduced to pounds by multiplying by 60, and pounds into tons by 
dividmg by 2000. Thus you will have the cubic feet, the pounds, 
and the tons, which the stream supplies every second. 

Note. Here, as in other cases, I adopt the ton of the revised 
laws of the State of New. York. Students have only to apply com- 
mon sense, when they have occasion to adopt the gross ton (2240) 
and, consequently, corresponding numbers in other weights. 

Sec. 307. It frequently happens, that a dam is built, and works 
are already in operation ; but the whole of the water is not employ-. 



140 

ed. Additional works are to be added ; and the engineer is called 
upon to estimate the quantity of waste-water, which pitches over the 
dam. The word Weir, or Waste-weir, is applied to this water- 
pitch, because such a pitch is in use for passing off the excess of 
water, in freshets, from canals, &c. It takes its name from the 
strong wires (weirs in German) which are inserted, to prevent inju- 
ries, which might occur by drifting over small articles of value. 
Directions for calculating the quantity of water in cubic feet, which 
crosses the weir per minute, are given in the next section. 

Sec. 308. Take the depth of the sheet of water by setting a very 
thin scale with a sharp edge against the stream, just touching the 
extreme edge of the waste-board. This measure will be sufficient 
if the said edge is perfectly horizontal throughout the width of the 
sheet of water. But the sheet must be divided into sections, and 
they measured separately, for each change in its level. Take the 
measure of the whole breadth of the dam, as well as of each sec- 
tion which you may think it necesary to make. Take the cubic 
feet per minute, set in the table against the inch of depth. This 
would be the true answer required, if the sheet of water was but one 
inch wide and confined by side-boards or walls. But if the sheet 
of water is 50 feet wide, or of any other width more than an inch, 
multiply the said cubic feet which pass down in the inch sheet per 
minute, by the whole width of the sheet, taken in inches. This 
would give the required answer, were the flow of water the same 
in a confined situation, as when flowing freely over a broad space. 
To compensate for this difference, divide the above product by 20, 
and add the quotient ; which will give the true answer. 

Sec. 309. If the measure of the depth is found in inches and 
quarters of inches, take the whole in quarters. As for 2 inches and 
3 quarters, look for the cubic feet against 11 inches (as this is the 
number of quarter inches) and take the eighth part of the said cubic 
feet, set against 11 inches. 

Sec. 310. If the table of depths is not sufficient for the measure 
of the depth of the sheet, take such one of the depths as will pro- 
duce the measure by doubling, tripling, quadi'upling, &c., and mul- 
tiply the cubic feet set against the taken depth, by the formula set 
against said doubling, tripling, <Sz;c., so assumed. Then divide said 
product by 20, -^.nd add the quotient, as in other cases. By this me- 



141 

tiiod the largest rivers, which fall down perpendicular rocks, may 
be calculated : and it is the best method in all water pitches. 

Sec. 311. Table for calculating Weirs, or sheets of water falling 
over dams, overfalls, or upon over-shot wheels. 









Formula of 


Depth of the 


Cubic feet of 


Number of 


numbers set a- 


sheet of water 


water per min- 


times for doub- 


gainst each doub- 


to be calculated ; 


ute ; discharged 


ling, trippling, 


ling, tripling, &c. 


taken precisely 


when the sheet 


&c., as referred 


to be used as mul- 


at the edge of the 


of water is one 


to in the last sec- 


tipliers, set forth 


plunge. 


inch wide. 


tion. 


in the last sec- 
tion. 


1 


0.428 


Twice taken. 


2.828 


2 


1.211 


Three times. 


5.196 


3 


2.226 


Four times. 


8.000 


4 


3.427 


Five times. 


11.180 


5 


4,789 


Six times. 


14.697 


6 


6.295 


Seven times. 


18.520 


7 


7.933 


Eight times. 


22.627 


8 


9.692 


Nine times. 


27.000 


9 


11.564 


Ten times. 


31.623 


10 


13.535 






11 


15.632 






12 


17.805 






13 


20.076 






14 


22.437 




- 


15 


24.883 






16 


27.413 






17 


30.024 






18 


32.710 







Note. This table and the preceding directions for its use, sup- 
pose the pond above the fall to be as nearly stagnant as it can be, 
when it merely gives motion to the sheet of water. If the water 
reaches the fall with any material degree of velocity, proceed thus : 
1st. Calculate the area in feet of the water-sheet at the pitch, by 
multiplying its depth by its width. 2d. Find the distance in feet 
which the water above the dam flows per minute, in the usual way, 
by throwing in floating bodies. 3d. Multiply said distance by said 
area ; and add the product to the cubic feet obtained by the appli- 
cation of the table, as before directed. This sum will be the cubic 



142 

feet of waste-water which pitches over the dam, or weir-board, per 
minute. 

Sec. 312. The descent of water to a lower level is all that is to be 
estimated in calculating its power^ After the supply of water aiford- 
ed by a stream is estimated, its power in driving machinery is to be 
calculated by applying the laws of Hydrodynamics, as set forth in 
sections 136 — 140, referred to in section 299 of this article. 

Sec. 313. Directions for calculating the efficiency of water-power 
as issuing from the side of the bottom of a flume. Measure in feet 
and decimals, the height of the point intended for the uniform level 
at the top of the flume, above the central point of the place intended 
for the gate-hole. Extract the square root of that measure. Mul- 
tiply 8 (the length of the horizontal jet per second under 1 foot head) 
by the said root ; which product will be the length of the jet of 
water per second at said gate-hole — its velocity to be estimated at 
the precise end of the contraction of the vein ; about the reduction 
of one third. Find the area of the intended gate-hole in feet and de- 
cimals ; and (after deducting for contraction of vein) multiply it by 
the length of the jet ; which will give the cubic feet issuing per se- 
cond. This will furnish the true measure of the water to be taken 
per second from the calculated supply of the stream. 

Sec 314. Each cubic foot of water, taken at the top of the 
stream, weighs 60 ib. But when it issues as a jet, or spouting fluid, 
from under a considerable head, it strikes with a force far exceed- 
ing its mere steelyard weight. Take this example for an illustra- 
tion of the principle explained in section 140. Falling bodies, as a 
cubic foot of ice weighing 60 ife., increase their velocity as the 
square root of the distance fallen. Beginning with 60 ife. (a cubic 
foot ice-cake) and suppose it merely pressing with 60 ft. weight. 
In falling (suppose in a vacuum) its acquired velocity is 8 feet per 
second at the end of one foot. In falling 9 feet its acquired velo- 
city is 24 feet per second; striking with the weight of 1440 ife. per 
second, if such blocks follow each other instantaneously. Its velo- 
city gives an increased impetus against whatever it strikes, accord- 
ing to the increased velocity 'of each block. As the steelyard weight . 
is merely 60 ife. pressure, without any acquired velocity, its impetus 
will be as the square of the acquired velocity. 

Sec. 315. In turning an undershot wheel, without any load, the 



143 

periphery of the wheel will move just as fast as the water. For 
example ; as water, issuing from a flume of 16 feet head, will move 
with a velocity of 32 feet per second (see sec. 140) a wheel of 10 
feet diameter will perform a revolution every second, and a small 
fraction over. The velocity which a given head will produce with 
the periphery of a wheel, where no friction nor power required to 
move machinery are allowed for, can be calculated in a very sim- 
ple manner. Find the velocity of the given head by section 140. 
Calculate the quantity of water, in cubic feet, effused at the gate- 
hole per second, after allowing for the contraction of the vein. 
Reduce the cubic feet to pounds. Then you have the velocity of 
the rim of the wheel, and of its effective power in pounds, for each 
second ; but nothing is yet allowed for friction or load. This must 
depend solely on careful trials. 

Sec. 316. The improvements made in machinery, since the time 
that Evans made his trials and calculations, are so great, that I be- 
lieve I shall proceed best in the object of this treatise, by making 
some calculations upon facts learned at the Poestenkill mills in this 
city (Troy, New- York.) I shall leave the comparison, as to effi- 
ciency, between undershot, breast, and overshot wheels, to mill- 
wrights. It is my opinion from my own observations (but they have 
not been extensive ; and I do not offer my opinion as a shadow of 
authority on this head,) that the horizontal submersed wheels are 
the best of the undershot kind. I shall treat the subject as if the 
three modes were equal, if the workmanship is equal. 

Sec. 317. Standard of the efficiency of water-power, taken from 
Poestenkill flouring mills, in Troy, New- York. 

Three flouring mills on Poestenkill, called Troy mill. Canal mill, 
and Globe mill, have been recently rebuilt upon the most approved 
model for overshot mills. The water-wheels of the Troy mill and 
Canal mill are 18 feet in diameter — with buckets 17|- feet long ; 
and the water pitches upon each from but one foot above. Each 
turns four run of stones. These are the property of Mr. Phineas H. 
Buckley. The Globe mill has its water-wheel 12 feet in diame- 
ter, with buckets 20 feet long ; and the water pitches upon the 
wheel from about two feet above. This wheel turns five run of 
stones. It is the property of Messrs. Vail and Townsend. The 



144 

three mills are near each other, and are driven by the same water, 
taken in succession. 

Sec. 318. The Troy mill and the Canal mill are constructed so 
nearly similar, that the description of either applies to both. This 
day, Feb. 13, 1838, I examined one of the mills, assisted by W. Lap- 
ham, a student ; and received full explanations of all we desired 
from the proprietor and his experienced miller. When the water 
is at a middling height, if but two run of stones are driven, each 
grinds 250 bushels of wheat in 24 hours — that is, a little more than 
10 bushels per hour. By careful measure and two calculations, we 
find that 40 tons of water strike the wheel per minute, or two thirds 
of a ton per second. The wheel revolves four times per minute,, 
and each of the stones, which are 4 feet 9 inches in diameter, re^ 
volves 146 times per minute. Either two run of either mill, grind 
500 bushels in 24 hours ; but when the whole 4 run are in action 
in either mill they will all grind but 700 or 800. 

Sec. 319. The Globe mill, with its 12 feet wheel and 20 feet 
buckets, grind 250 bushels in 24 hours with one run ; but I under- 
stood, that it could scarcely grind 500 bushels with two run at once. 
But that the whole five run would grind about as much in 24 hours, 
as the four run in either of the other mills. 

I shall not describe the geering any farther, than to say, that they 
have no primary cog-wheels on the water-wheel shafts. Cogs are 
set upon the end of the perimeters of the water-wheels, and mesh 
into small cog-wheels on transferring shafts. These shafts and 
cog-wheels transfer the power of the water-wheels to the cog-wheels 
which turn the spindles. 

Sec 320. I will now attempt to make some applications of Evans' 
rules to the preceding facts. After a calculation has been made by 
the engineer of the supply of water, and he has taken a measure of 
the fall, &c., he will compare the efficiency of the mill-seat, by ex- 
tracting the square root of his water head, and compare it with the 
square root of the head at these mills. The rule is the same, in the 
application of the square root of the height of the water, as illustrated 
in sections 137 to 140 ; always having in view, the supply. 

Sec 321. Calculations for efficiency and supply of water. Evans 
says, that for grinding 3.8 bushels per hour, 36.582 square feet of 
the face of the stones must pass over each other per minute. In 



145 

pursuance of his mode of estimating, I have made the following cal- 
culations. The dressed faces of the stones of the Poestenkill mills 
are 17.278 square feet to each. Therefore, at every revolution of 
the stones, 298.5 square feet of them rub over each other. As the 
stones revolve 146 times in a minute, it follows that 43.585 square 
feet receive the triturating rub per minute. And 10 bushels are 
ground per hour ; therefore something more than two million and a 
half square feet of mill-stone face (2.615.100) are applied to grind 
10 bushels. And as 40 tons of water are expended upon the wheel 
per minute while driving two run of stones in grinding 10 bushels 
each per hour, this is the clear result : 2400 tons of water, driving 
stones to rub each other's faces over surfaces of about five and a 
quarter million feet (5.230.200) will grind 20 bushels of wheat in 
one hour, with two run of stones, driven by one wheel. 

Sec. 322. If the Poestenkill mills may be taken as standards of 
reference, we may apply these abridged data : 40 tons of water ap- 
plied per minute to an 18 feet wheel with 17^ feet buckets, will drive 
two run of stones to triturate about five million square feet of faces in 
grinding 20 bushels of wheat, in one hour. From these data smaller 
or larger streams and fallsmay be estimated ; always having refer- 
ence to the proportional principle of square root, as explained in 
sections 137 to 140. 

Sec. 323. To understand this example for calculating the supply 
of water on the weir method, when pitching from the termination of 
an apron, or over a weir-board, see sec. 307 to 311. 

The first example was calculated from measurements taken of 
the depth of the sheet of water pitching upon the wheel, depth 5^ 
inches, and width 17| feet. As 5.5 inches have no formula in the 
table, 22 quarters must be taken, which is beyond the table. There- 

19 



146 

fore the 22 quarters must be made out according to the rule given 
in sections 309, 310. This gives the 

5.652162 as the cubic feet for 1 inch breadth. 
209 width of the sheet in inches. 



50869458 
113043240 



20)1181.301858 

59.065 the twentieth to be added. 



1240.366 cubic feet. 

60 pounds in weight per cubic foot. 



2000)74421.960(37.21 tons per minute. 
6000 



14421 ^ 

14000 



4219 
4000 

2196 
2000 



196 



Sec. 324. For several reasons, I was not satisfied that 5^ inches 
was the true depth of the sheet ; therefore the following calculation 
was made, on the assumption that the sheet of water was 6 inches 
deep and 17^ feet wide — or nearly so. 



147 

According to the table, 6 inches depth gives, in a sheet of 1 inch 
width, 6.295 cubic inches per minute. 

6.295 cubic feet for an inch in width. 
209 width of the sheet in inches. 



56655 
125900 

20)1315.655 

65.782 twentieth to be added. 



1381.437 cubic feet. 

60 pounds weight of cubic feet. 



2000)82886.220(41,44 tons per minute. 

8000 37.21 tons per minute, sec. 323. 



2886 4.23 difference. 

2000 



8862 40 tons per minute the best average 

8000 for all the wheels. 



8622 
8000 

622 

Sec. 325. Examples of a calculation of the triturating surfaces of 
mill-stones. The stones of Poestenkill mills are 4 feet 9 inches in 
diameter, a nine-inch space in the centre is allowed for the spindle, 
&c., unfaced. The mean circle is 8.639 feet. The breadth, out- 
side of the nine-inch area of the centre, is 2 feet. The 2 feet mul- 
tipUed into the mean circle (8.639) gives 17.278 as the area of the 
triturating superficies of the stone. 

Sec. 326. The triturating superficies of the stone is to be multi- 
plied by itself; for every hair-breadth starting of the stone gives 
friction for all its superficies. Therefore 17.278 is to be multiplied 
by itself, to give the superficial friction of one revolution. It must 
then be multiplied by its revolution per minute. In this case there 
are 146 revolutions per minute. Therefore the product of 17.278 



148 

multipled by itself (producing ,298.5) must be multiplied by 146. 
This gives 43585 per minute. 

Example of a calculation for the surface of collision wliich is ne- 
cessary between mill-stones, for grinding 20 bushels per hour. 

The faced surface of each mill-stone, used in the Poestenkill mills, 
being 17.278 feet, it must be multiplied by 17.278, to produce the 
superficial measure of surface in the act of friction. As this multi- 
plication gives the surface of friction, or trituration, at but one revo- 
lution, this product must be multiplied by the number of revolutions 
as aforesaid, per minute (146 times in this case.) The operation 
is, therefore, as follows : 

17.278 the superficial area of the stone. 

17.278 



138224 
120946 
34556 
120946 

17273 

298.528284 surfece of trituration for one revolution. 
146 revolutions per minute. 



1791168 [reject three last decimals.] 
1194112 

298528 



43585.088 area of the trituration per minute. 
60 minutes per hour. 



2.615.105.28 area of trituration per hour in grinding 10 bush- 
els on one run of stones. 
Or, 5.230.210.56 area for 20 bushels, by this force of water, on 
two run of stones, driven by one wheel. 

TOPOGRAPHY. 

Sec. 327. Topography* is the science and art of locating defi- 
nitely, and describing accurately, any district or limited portion of 

* Topography {topos, p\ace, grapjie, description, Greek) is applied, literally, to any spot of 
earth, great or small ; but it is mostly applied to the description of a large district of country. 



149 

territory, for the purpose of effecting a particular object. It may be 
applied to the object of locating a fortification — of locating a propos- 
ed city — of locating (on a map) rivers, mountains, rock-strata, coal 
beds, marble quarries, &c. This article will be confined to the ma- 
thematical operations required in all cases of accurate topography. 
Students in engineering must be exercised in a kind of miniature 
series of operations ; and this will be sufficient to qualify an ordi- 
nary genius for extensive undertakings. The principle being the 
same, enlarging a plan will not embarrass the correct scholar, who 
has carefully gone through a small-scale process. 

Sec. 328. After studying, efficiently, the preceding rules and di- 
rections, nothing remains but additional applications. A case in 
practice, historically given, is preferred to series of rules. Such 
cases are within the limit of any school. 

The students of Rensselaer Institute undertook (several classes in 
succession, during the last four years) to settle the relative topo- 
graphy of Durham Peak on Catskill mountain, the Rensselaer In- 
stitute in Troy, and the Wiswall house on the west side of the Hud- 
son river. The first step [extemporaneous step) was to settle the 
proximating height, course, distance, &c., by the barometer, sex- 
tant, and compass. They used the barometer at each station as 
described in sec. 168 — the sextant as described under instruments, 
and under latitude and longitude — the compass as described under 
surveying. The results of these observations they plotted, as a 
guide for accurate observations ; taking the differences of latitude 
and longitude for measured lines. 

Sec. 329. They commenced the series oi definite ohservations by 
assuming a suitable location for a base line — they took Fourth-street, 
in Troy, because it was accurately levelled about a mile in length. 
The height of the street above the tide-water level of the Hudson 
river was taken. At each end of the measure on Fourth-street, the 
bearings were taken by the compass, to the Wiswall house and to 
the Rensselaer Institute. The angular distances were also taken 
with the sextant, from monuments on each end of Fourth-street. It 
must here be remarked, that the slits in the sights of the compass 
gave the angles at the Institute and Wiswall's on the level with 
Fourth-street ; but that the sextant gave the angles on the principle 
of an oblique plane to them from Fourth-street — of course the latter 



150 

angles would be smaller than the former, and proportionably larger 
on the base line. Common sense will suggest the results of calcu- 
lations on each. 

Sec. 330. The length of the base line from the Institute to Wis- 
wall's, could be obtained without further direction, by any student, 
who had studied this treatise so far, with attention. Having this base 
line, the distance to Durham Peak was readily obtained. The bear- 
ing of Catskill mountain from the Institute and from Wiswall's, was 
taken by the compass, and the angles in both places by the sextant. 
The level to the mountain was also taken at both places, and the 
angular height above the levelled point on the mountain. The 
height was calculated by the base line to the mountain, and the an- 
gle. The descent to tide-water level, below the point of levelling, 
was taken by calculating the convexity of the earths 

Sec. 331. The height of the Institute and of Wiswall's above 
tide-water were taken by the level and rod ; the barometer being 
considered but an approximating instrument, well adapted to an ex- 
temporaneous survey. With these data, any student in the pre- 
ceding part of this treatise, can make out all that is required. And 
he will be able to use the distance from Wiswall's to Durham Peak, 
as a base line to find the distance to Saddle mountain, near Wil- 
liams College, in Massachusetts. Then he could use the distance 
from Saddle mountain to Catskill mountain for a base line to find the 
distance to Beacon mountain in the Highlands on Hudson river. 
Thus he might go on indefinitely, and settle the topography where- 
ever he chose. But, as errors of small beginning will increase in a 
fearful ratio, a base line of one mile cannot be relied on for such 
extensive measures. 

Sec. 332. Having illustrated the principle sufficiently for the 
comprehension of any correct student, I will now extend the same 
method to settling the topography of whole States or of any other 
territory of ever so great extent. More exact instruments are 
always required in extensive surveys ; as errors will increase by 
extending lines, which are erroneous at the outset in a small degree. 
A theodolite, or other telescopic levels and graduated instruments, 
must be employed in all cases, where the reflecting quadrant or 
sextant are not used. For no ordinary sighting instrument is sufii- 
ciently accurate. I say I will extend the method ; but I do not 



151 

mean to extend my directions any farther than to say, that all the 
preceding directions require enlarged measurements, greater care, 
frequent reviews, and telescopic insti:uments. 

Sec. 333. For extensive topographical surveys, several base 
lines of great extent ought to be established. Take for example, 
the State of New- York. A base line might be surveyed on the ice 
in the Hudson river, of 50 or 60 miles in length ; and monuments 
erected on shore for each mile, carefully noting the precise distance 
at right angles from the place of the line on the ice. Other base 
lines might be established along the shores of frozen lakes, and in 
such extensive plains as Schoharie Flatts, the 70 mile canal level, 
&;c., &c. These might be referred to for correcting trigonometri- 
cal surveys deduced from d'ifferent bases. 

Sec. 334. In taking a topographical survey, much more depends 
on the ingenuity and faithfulness of the surveyor, than on any pa- 
rade of outfit, or train of subordmates and assistants. One good 
iJieodolite of the best modern structure, . a good sextant, a telescopic 
level, a sea-telescope for taking longitude by Jupiter's satellites, and 
a well-tried pocket-chronometer, will be sufficient for a practical 
mathematician, who does not make his employment " a sinecure." 
But faithful and intelligent flag-men, rod-men, and chain or tape- 
bearers, I have always found to be of more importance than a host 
of assistants, crowded into the corps to gratify influential relatives. 

MATERIALS FOR CONSTRUCTION. 

Sec 335. The world is full of books on this subject. It is rather 
a reading subject than a mathematical one. Any one can sit in his 
closet and write (rather compile) a treatise on materials for con- 
struction. Though this treatise is prepared solely for practice, ma- 
terials for construction in general come within its object. For a read- 
ing book, Professor Mahan's treatise is recommended as a com- 
pend of every thing worth reading on that subject.* I propose but 
few materials on this vastly extended subject ; and these are intend- 
ed to be adapted to the discipline of students in mode of thinking on 
the subject, rather than to the theory and practice. 

* It is a matter of regret that the title of tliis excellent treatise on materials for construc- 
tion, was 60 inappropriately selected. 



152 

Sec. 336. Materials for construction are very naturally divided 
into Inorganic and Oeganic. The Inorganic are those which are 
governed by the laws of affinity, uninfluenced by the living princi- 
ple — as rocks, earths, and metals. Organic are those which re- 
ceived their forms or structure from the unexplained action of the 
living principle — as timber, bones, teeth, &c. As they owe their 
origin to a forced structure, induced by the living principle, they 
are predisposed to a departure from their present structure ; conse- 
quently are less durable. 

Inorganic Materials foe, Constexjction. 

Sec. 337. These are subject to decay on two principles — decom' 
position and disintegration. Decomposition is the separation of con* 
stituent atoms, by the interference of extraneous atoms which, by 
force of chemical attraction, draw asunder previously combined 
atoms. Carbonic acid, for example, unites with atoms of iron, se- 
parating them from the general mass, and forming the yellowish 
iron-rust. Oxygen also unites with atoms of iron, forming the red 
iron-rust. Disintegration separates whole molecules of masses, 
without separating their constituent atoms ; as the crumbling down 
of a rock of slate, of limestone, of graywacke, &c. In such cases 
the smallest fragment, or molecule, is but a lessened rock, retaining 
its original atomic constituents. 

Sec. 338. Most persons know the names oithnler, as oak, pine, 
cedar, &c., at sight ; and need no direction on that subject. Few 
know the names of rocks and earths ; and need some instruction on 
the characteristics by which they are known. As they are pretty 
accurately distributed into strata by modern geologists, I shall give 
a few concise directions by which they are known. I consider this 
the more important, as canal and rail-road reports recently include 
geological characters. 

Geological Alphabet. 

Sec. 339, Rocks are mostly made up of aggregations of homo- 
geneous minerals ; in some cases a rock consists wholly of a single 
homogeneons mineral — as limestone, argillite, &c. The annexed 
Geological AlpJiahet must be learned by an inspection of the speci- 
mens only ; no description can convey an adequate idea of them. 



153 

Sec. 340. Every stratum of rock, or earth, consists of one or 
more of these nine minerals ; therefore they are aptly denominated 
the Geological Alphabet. They are— 1. Quartz, 2. Felspar, 3. 
Mica, 4. Talc, 5. Hornblende, 6. Argillite, 7. Limestone, 8. Gyp. 
sum, 9. Chlorite. 

Quartz. When held between the eye and a window, it reflects 
light somewhat like a polished piece of cold tallow or glass. This 
is called its lustre. On attempting to scratch it with a pen-knife, 
the metal will leave a trace on it, and it will not be scratched. It 
is commonly glass color, but it is often milk-white, reddish, and of 
various other colors. Quartz consists of about 93 per cent silex, 6 
alumin or clay, 1 lime, besides 2 or 3 per cent of water in a solid 
state. 

Felspar, or Feldspar. Its lustre is peculiar ; but it in some mea- 
sure resembles that of a broken edige of china-ware. It may be 
scratched with a knife. Its color is generally white or flesh-color- 
ed. It is best ascertained when in the state of a rock aggregate, by 
procuring an outside fragment, which had been long exposed to air 
and moisture. In this state it always assumes a peculiar tarnish, of 
a dirty yellowish hue. Felspar consists of about 63 per cent silex, 
17 alumin, 13 potash, 3 lime, 1 iron, 3 water. 

Miea. It is always in shining lamina or scales. It is every 
where known by the improper name of isinglass. The scales are 
always elastic. It consists of 48 per cent silex, 34 alumin, 9 potash, 
5 iron, 1 manganese, 3 water. Black mica contains 22 per cent 



iron. 



Talc. It often resembles mica ; but can be distinguished from it 
by being non-elastic. Take a small scale or fibre of it in a pair of 
tweezers, put it under a magnifier, and bend it with the point of a 
fine needle. If it remains as it is bent, it is talc ; if it springs back 
it is mica. It always gives a rock the unctuous or soapy feel. It 
consists of 62 per cent silex, 27 magnesia, 2 alumin, 3 iron, 6 
water. 

Hornblende. It is the toughest of all earthy minerals. Gene- 
rally it presents a kind of confused fibrous structure. ' It may be 
scratched with the knife. The color is always greenish, brownish, 
or black. Sometimes it appears in black scales, resembling mica to 
Jhe naked eye ; but under the magnifier it differs materially. It 

20 



154 

consists of about 42 per cent silex, 12 alumin, 11 lime, 3 magnesia, 
30 iron, 1 manganese, 1 water. It is very heavy. 

Argillite. This needs no description. The common roof-slate, 
and the slate used for cyphering on in schools, are good specimens. 
It consists of about 38 per cent silex, 26 alumin, 8 magnesia, 4 lime, 
14 iron, 10 of potash, soda, manganese, water, &c. 

Limestone. Common marble affords good specimens. It should 
be tested by a drop of muriatic, nitric, or sulphuric acid, which will 
cause an effervescence or bubbling. It consists of 57 per cent of 
lime, 43 of carbonic acid. Sometimes it is colored with iron ; and 
often contains a little silex and alumin. 

Gypsum. Common plaster of Paris. It will not effervesce with 
acids, and is generally softer than limestone. It consists of 32 per 
cent lime, 46 sulphuric acid, 22 water. 

Chlorite. It is a little harder than talc, but may be scratched with 
the finger nail. Under the magnifier it appears like a compact 
mass of fine green scales. When breathed on it gives an odour in 
some measure resembling clay. Its elementary constituents are 
variable. They will average about 40 per cent silex, 23 alumin, 
18 magnesia, 15 oxid of iron, 2 lime, 2 water. This is the least 
important of the whole nine. 

After students can readily recognise these nine minerals, they 
should be exercised in pointing them out in their various states of 
aggregation. They will soon be enabled to spell out any rock with 
facility. 

Arrangement of Rocks for the Instruction of Engineers. 

Sec. 341. The science of Geology consists in a systematic ar- 
rangement of facts, explaining the structure of the earth. 

Our observations are limited to its exterior rind or coats. We 
know very little of its interior structure. But the inequalities of 
its surface often give us admission to a considerable depth ; from 
which we should be totally excluded were its surface every where 
smooth like a Pacific sea. 

Geology teaches us that minerals which are associated in one dis- 
trict of country, are associated in the same order in all other dis- 
tricts. Hence the experience of the miner and the quarry-man in 



155 

any country, may be applied in searching for useful minerals in all 
other countries ; for geology is the true science of mining. 

Five classes or series of Deposites. 

We have five series of strata, or a five-fold repetition of a carbo- 
niferous, quartzose, and calcareous fornaation. 

First series (primitive) is terminated by granular limerock, desti- 
tute of any organized remains. 

Second series (transition) is terminated by compact and sJielly 
limerock; the compact perforated with encrinites. 

Third series (lower secondary) is terminated by corniferous or 
cherty limerock. It contains horn-stone and abounds in stone horns. 

Fourth series (upper secondary) is terminated by oolite or coral 
rag, a calcareous stratum containing corals. 

Fifth series (tertiary) is earthy, and terminated by a lime deposit 
of shell-marl. 

To these may be added the red sandstone group, or the gypsum 
and salt associates ; as along the Erie canal, between Utica and 
Lockport. Also the cretaceous deposite ; as the green-sand marie 
of New Jersey, &c. 



156 

Exhibition of Geological Strata. 
[From the Geological Text-Book.] 

Sec. 342. Fig. 3. This figure represents a segment of the earthy 
from the Atlantic to the Pacific, between the 42° and 43° N. lati- 
tude, in its present state, so far as regards rock strata. 

Abbreviations. Car. Carboniferous formations ; Qu. Quartzose 
formations ; Cal. Calcareous formations. The numerals indicate 
the first, second, third, fourth, and fifth series of formations. The 
fifth, however, is not represented here. 

According to the modern theory of the earth, these strata of rocks 
were deposited in concentric hollow spheres, like the coats of an 
onion. And within the granitic sphere, combustible materials were 
deposited ; and all strata above were broken up in several north 
and south rents, by their combustion and explosion. For example, 
one passed through New England, the Highlands, Virginia, &c. ; 
another included the Rocky Mountains, Andes, &c., as here exhi» 
bited. 



157 




SadsojtJt* 
Green Jilh* 



15S 

Descriptions of Strata. 

Sec. 343. Class I. Primitive or First Series. 

1. Carboniferous or sJaiy formation. 

Granite, is an aggregate of angular masses of quartz, felspar^ 
and mica. Subdivisions. It is called crystalline (granite proper) 
when the felspar and quartz present an irregular crystalline, not a 
slaty, form. It is called slaty (gneiss) when the mica is so interpos- 
ed in layers as to present a slaty form. 

Mica-Slate, is an aggregate of grains of quai'tz and scales of 
mica. 

Hornblende Rock, is an aggregate, not basaltic, consisting 
wholly, or in part, of hornblende and felspar. 

Talcose Slate, is an aggregate of grains of quartz and scales of 
mica and talc. Subdivisions. Compact, having the laminae so closely 
united that a transverse section may be wrought with a smooth 
face. When the quartzose particles are very minute and in a large 
proportion, it is manufactured into scythe-whetstones, called Quin- 
nebog stones. Fissile, when the laminse separate readily by a blow 
upon the surface. Varieties. Cliloritic, when colored green by 
chlorite. It contains gold in the Carolinas, and probably through- 
out its whole range by way of New- York, to Canada. 

2. Quartzose formation. 

Granular Quartz, consists of grains of quartz united without 
cement. 

3. Calcareous formation. 

Granular Limestone, consists of glimmering grains of carbo- ' 
nate of lime united without cement. Dolomite, when it consists in 
part of magnesia, and is friable. 

Sec. 344. Class II. Transition or Second Series. 

1. Carloniferous or slaty formation. 

Argillite, is a slate rock of an aluminous character, and nearly 
homogeneous, always consisting of tables or laminae whose direction 



159 

forms a large angle with the general direction of the rock. Clay 
Slate, when the argillite is nearly destitute of all grittiness, and con- 
tians no scales of mica or talc. Wacke Slate, when it is somewhat 
gritty and contains glimmering scales of mica or talc. Roof Slate, 
when the slate is susceptible of division into pieces suitable for roof- 
ing houses, and for cyphering slate. 

2. Quartzose formation. 

First Graywacke, is an aggregate of angular grains of quart, 
zose sand, united by an argillaceous cement, apparently disintegrated 
clay slate, spangled with glimmering seal es. Millstone grit and grey 
ruhble,* when the grains are in part coarse, and more or less con. 
glomerate, either white or grey, often very hard. 

3. Calcareous formation. 

Sparry Limerock, consists of carbonate of lime, intermediate in 
texture between granular and compact ; and is traversed by veins 
of calcareous spar. 

Calciferous Sandrock, consists of fine grains of quai-tzose sand 
and of carbonate of lime, united without cement, or with an exceed, 
ing small proportion. 

Metalliferous Limerock, consists of carbonate of lime in a ho- 
mogeneous state, or in the state of petrifications. Birdseye marlle, 
when the natural layers are pierced transversely with cylindric pe- 
trifactions, so as to give the birdseye appearance when polished. 

Sec. 345. Class III. Lower Seconbary or Third Series. 
1. Carboniferous or slaty formation. 

Second Graywacke, is an aggregate of grains of quartzose sand, 
less angular than those of first graywacke, and generally contains 
some fine grains of limestone. 

It is sometimes gritty, and contains a few glimmering scales ; 
but it is often a soft slate and dark brown. It rests upon the 

* Rtiible being an uncouth Word, but too well established to bo rejected, I will state : that 
in common English it signifies a hard grey stone, in roads, of a spheroidal form, which causes 
the rumbling and jolting of carriages. Kirwan calls these stones, common gray wacke, as 
opposed to graywacke elate. 



160 

shelly kind of transition limerock, and is the lowest of our secondary 
strata. 

2. Quartzose formation. 

Millstone Gkit and Rubble, are composed of quartzose peb- 
bles and grains cemented together ; often very hard. 

Remark. A distinct stratum called old red sandstone, and another 
called millstone grit, have been given by most geologists. But the 
latest European geologists very properly reject them. Because the 
red sandstone is found passing into all the three graywackes. 

3. Calcareous formation. 

Geodiferofs LrMEEOCK, consists of carbonate of lime, combined 
with a small proportion of argillite or quartz in a compact state, 
mostly fetid, and always containing numerous geodes. 

CoRNiFEROtrs Limerock, consists of carbonate of lime, embracing 
hornstone, and numerous species of petrifactions, called stonehorns 
(Cyathopyllum.) This stratum is called carboniferous limestone by 
Conybeare. 

Sec. 346. Class IV. Upper Secondary or Fourth Series.- 
1. Carloniferous or slaty formation. 

Third Graywacke, is an aggregate of grains of quartzose sand 
and pebbles, less angular than those of first and second graywackes,, 
and generally contains fine grains of limestone. 

It is sometimes gritty ; but often soft. It rests on the carboni- 
ferous limerock of foreign geologists, who often call it grit slate. 

2. Quartzose formation. 

Millstone Grit and Rubble, are composed of quartzose peb- 
bles and grains, cemented together ; often very hard, 

3. Calcareous formation. 

Oolitic Rocks, are aggregates, which contain more or less of 
carbonate of lime of an earthy texture, either compact, or granut 
lated. 



161 



Sec. 347. Class V. Tertiary or Fifth Series. 
1. Carboniferous formation. 

Plastic Clay, that kind of clay, generally called potter-baker's 
clay, which will not effervesce with acids. When it is white it is 
called pipe-clay. 

Marly Clay, that kind of clay which will effervesce with strong 
acids. This stratum is almost universal. 

Marine Sand and Ceag. The sand consists of fine grains of 
quartz, not united by adhesion or cement, but in loose masses which 
may mostly be poured. The crag consists of pebbles, clay and 
loam, either united by carbonate of lime or iron cement, as pudding, 
stone ; by clay and iron cement, as the hard-pan ; or not united, 
being merely stratified gravel ; or united by adhesion, as the arena, 
ceous concretions near Troy, on Green Island. The marine sand 
occupies a broad strip on the west side of Hudson river, from near 
Lake Champlain to a distance of 100 miles. 

Shell-Marl, is' in insulated or continued layers, fields, or patches, 
in almost every part of the earth. It consists chiefly of broken, pul- 
verized, and entire shells, of the genus helix (genera helix, planorhis 
and lymnea of Lamarck.) 

Sec 348. Subordinate Series embraced in the Third Regular 

Series. 

(Lower Secondary.) 

1. Carboniferous and quartzose formation. 

Saliterous Rock, consists of red, or bluish-grey, sand or clay. 
marl, or both. In some localities they form the floor of salt mines 
and salt springs. 

2. Quartzose and slaty farTnations. 

Ferriferous Rock, is a soft, slaty, argillaceous, or a hard, sandy, 
siUceous rock, embracing red argillaceous iron ore. 

3. Calcareous formation. 

LiASOiD, is an argillaceous limestone, with an admixture of mag- 
nesia, iron, and finely pulverized quartz ; forming a compound of 

21 



162 

homogeneous aspect. On burning and mixing as in the manufac- 
ture of mason's mortar, it becoirjes a solid cement under water. 

Sec. 349. Anomalous Deposits. 

1. Volcanic. 

Basalt, which is called Amygdaloid, when amorphous, of a close 
texture, but containing cellules, empty or filled. When the amyg- 
daloid has a warty appearance and resembles slag, it is called toad- 
stone. Columnar basalt, is presented in prismatic polygons more or 
less regular. 

Sec. 350. 1. Useful Rocks. 

Marble. This name is indefinitely applied to any rock of car- 
bonate of lime, which may be wrought into building stone with the 
chisel. The most valuable is the Granular limerock. In this coun- 
try it extends from Canada to Pennsylvania, along the west side of 
the Green Mountain range. Therefore, every farmer, whose lands 
lie in that range, may probably find on his farm at a greater or less 
depth, or at the surface, good statuary marble. Good marble has 
been found, called birdseye marble, in the compact transition lime- 
rocks ; also in the shelly kind. But shelly marble will readily dis- 
integrate on exposure to heat ; and in time, on exposure to the dis- 
integrating agents. 

Freestone. This name has usually been applied to red sandstone. 
But its application to all saliferous rocks (red, gray, or variegated) 
is authorised. The freestone of Rochester, on Genessee river, is 
red, gray, spotted, and variously colored. Saliferous rocks, and red 
wacke of the first, second, and third graywackes, make good free- 
stone, unless they contain pyrites. These stones never crack by 
heat; but they crumble off" or become friable, and ought not to be 
highly heated. 

Flagging-stones. The best known in this country are the gneiss, 
and gurtisseoid hornblende. Haddam, Conn., affords excellent spe- 
cimens. West of Worcester, Mass., towards, and in, Leicester, the 
rocks are equally good. They were formerly too far inland. 
Since the canal is made, they may become profitable. When I first 
heard of this project, I supposed the value of the stock was to be es- 



163 

timated by the value of these vast ledges of gneiss rock. Why- 
Yankee ingenuity and perseverance does not reach them, I know 
not. 

Wall-stones. This name is applied to stones which lie fairly in 
a wall, with or without chiseling or sawing. It is not necessary 
that they should bear chiseling, provided they can be broken from 
ledges in regular parallelepipeds for laying up in a dry wall, or for 
masoning with mortar. Gneiss, gneisseoid hornblende, and the two 
lower graywackes, afford more or less good wall-stone. First 
graywacke, when formative, is always a good wall-stone. It has 
been sawed and used in Troy with success, in the basement story 
of houses. See Dr. Gale's house. It is remarkable for cracking, 
and even flying into pieces, when heated ; hence it is called snap- 
stone in some districts. Third graj'^wacke is rarely suitable for a 
wall-stone ; for it contains iron pyrites, which often produces rapid 
disintegration. See some of the western locks, and the walls of some 
houses in Ithaca. 

Sec. 351. Millstone is a subdivision of graywacke. It is therefore 
first, second, and third. It is a good wall-stone, and will resist a great 
heat. Millstones were quarried from Shawangunk Mt. (first gray- 
wack) until the burrstone manufactories were extensively intro- 
duced ; and it was profitable to the manufacturers, and a public be- 
nefit. As they were wrought at Esopus (Kingston) in great quan- 
tities, they were long called Esopus millstones. Ledges of millstone 
grit, which would make tolerably good millstones, though liable to 
crumble, may be had in second graywacke near Utica, and in third 
graywacke on Alleghany mountains. 

Grindstone used in this country is a gray sandy variety of third 
graywacke. Two extensive layers of grindstone run nearly paral- 
lel to Schoharie Kill, in Blenheim, Schoharie county, on the estate 
of Judge Sutherland. One range is in the west bank of the Kill, the 
other is further west. From pieces of rock adhering to NovaScotia 
grindstones, I believe they are from the same rock. Numerous 
other localities of various qualities, are seen in the graywacke of 
Catskill and Alleghany mountains. 

Whetstones, called novaculite, are always a variety of talcose 
slate. When they are harsh they are called Quinnebog whetstones, 
or scythe whetstones. When they are fine-grained they are called 



164 

Turkey hone. They are wrought in Belchertown, Mass., and at 
Lake Memphremagog in Vermont. Though these whetstones are 
always a variety of talcose slate, they are found in this country at 
the nieeting of talcose slate with^mica slate or argillite ; when with 
the latter the whetstone is softer than when with the former. In 
Hawley, Mass., an inferior kind appears in connexion with the mi- 
caceous iron ore, at the meeting of the talcose slate and mica slate. 

Hones of a very excellent quality, are found in first graywacke 
near a place called the Red-Rockshire, in Columbia county, N. Y., 
and in Rensselaerville, Albany county, m third graywacke. I 
have seen layers of the same rock in numerous localities iii the third 
graywacke of Catskill and Alleghany mountain ranges. 

Sec. 352. Hornilende rocks are the toughest of all rocks ; con- 
sequently useful in fortifications, to resist the force of a cannonade. 
They are somewhat durable — those of the basaltic kind are best. 

Granular quartz and granular limestone are the most durable of 
all rocks ; and often present convenient forms or blocks, for use in 
building. 

Argillite, argillaceous grayivacke, saliferous slate, ferriferous slate, 
conchoidal lias, and pyritiferous slate, are subject to rapid disintegra- 
tion, and are not suitable for building materials. But they exceed 
all other materials for dams and other works, where the force of 
water is to be resisted. Hence the dam on the Hudson river in 
Troy, is made to resist the force of that mighty river by argillite 
taken from Mt. Olympus. 

Sec. 353. Cements made of lime and sand are most in use where 
walls are not continually exposed to water. The sand should be 
used immediately from the pit without drying. But if the original 
stone was made up of carbonate of hme and silicious sand, before 
it was burned, it makes a better cement for such exposed walls. 
This fact is mentioned by Glauber, who wrote two centuries ago. 
Recently, engineer Canvass White, Esq., has most efiectually re- 
vived the use of this cement in canal locks, &c. Oxyd of h'on or 
of manganese improves the cement. 

Gypsum cement. The gypsum is first ground at the plaster-mill ; 
then heated in a potash-kettle, or other iron vessel, for 24 or 36 
hours, to nearly a red heat. It is thus divested of its water of com- 
bination, being about 20 or 22 per cent. This dry powder is to be 



165 

kept from the air, by being enclosed in casks, until the moment it is 
to be applied. It is then wet with about its weight of water, for 
common use — more for nice plaster work — less for pointing and 
common mason work. A little glue is dissolved in the water, when 
some time is required for applying it, to prevent its hardening too 
soon. 

This cement is excellent for making a smooth finish, mouldings, 
&c., for inside walls. But it is not very durable on exposed sur- 
faces. The application must be made under the constant view of 
this fact : that its volume is a little enlarged after it has been appli- 
ed, by the farther absorption of water, or atmospheric vapor. 

Ti3iBEE Materials for Coa\structio:^. 

Sec. 354. Timber has always been used, and is necessary for cer- 
tain parts of stone or brick edifices — particularly for beams. And 
if timber is well selected, and well prepared and arranged, it will 
endure many centuries. The coffins of the Egyptian mummies, 
which are three thousand years old, are still perfectly sound. 

The white oak (quercus alba) has long been considered as com- 
bining solidity and durabilitj- in a pre-eminent degree. But it is 
much better in both respects, for having grown in open ground. 
And the best part of a tree is between the sap-wood (alburnum) and 
the heart ; as the heart is generally shaky, and the alburnum is not 
sufficiently solidified. Notwithstanding the durable character of 
the oak, it will not endure exposure to water, as well as cedar, and 
some other timber. It should be placed in situations which are se- 
cured from rains. It should be well seasoned under cover before 
it is used. If it is soaked in water before it is seasoned, it will be 
brittle and soon decay. For the water takes out all the soluble 
parts ; which, if dried in the timber, solidify it. 

Oak which grows in warm countries, is found to be the best. 
The soil should be neither wet or dry, and at about a medium in 
quality. For if it grows too thriftily or stintedly, it will be less 
solid. Soil, which contains considerable clay, produces better oak 
than a loose loamy or sandy soil. Very old or very young trees 
are not as good as those which are about 100 years old. 

Most of the preceding remarks will apply toother tunbers. It 
may be added, no timber should remain in the bark after being 



166 

felled. Neither should the sap-wood remain, if the timber is requir. 
ed for long duration. 

Sec. 355. Strong wood posts will sustain very great weights di- 
rectly in line with their fibres. But sawed posts, which have crooked 
grains, will be weakened by having their fibres cut off by the saw. 
No material can be employed which will retain its original length, 
through all temperatures and all degrees of humidity, so well as 
wood. 

Wood is very durable lolien placed in situations from loliicli rain 
is excluded ; and its decay is most rapid when it is alternately wet- 
ted and dried at short intervals. Hence the bottom of a vessel, 
which is always immersed in water, endures much longer than the 
sides (the parts between wind and water, in seamen's language) 
which are perpetually^subjected to the destructive effects of the sun 
and water. 

Direction of pressure. Carpenters should study to bring the 
greatest pressure as near the direction of the grains of the timber as 
possible. If a strong oaken piece of timber, one foot long, is sus- 
pended by one end, and has a hook at the other, we can scarcely 
conceive of a weight fastened to the hook sufficient to break it — its 
ends, however, must be supposed to be fastened by clamp-like bands. 

Depth of Beams. If the depth of a beam is great, its thickness 
may be small ; though a proportional thickness must be adapted to 
each particular case. All tapering timbers should have their ta- 
pering calculated upon the common lever principles. Spokes of 
carriage wheels, arms of mill wheels, &c., taper towards their ex- 
tremities upon the lever principle. 

Cross-grain timber. No honest carpenter will work in cross- 
grained timber, where strength is required, without giving his opi- 
nion of its insufficiency. In all such cases mere lateral adhesion of 
fibres gives all the strength. Some kinds of wood, such as the com- 
mon laurel {kahiia latifolia) are almost exceptions to the general 
rule. Still there is no timber which cannot be easiest divided in the 
direction of the fibres. 

Iron Materials for Construction. 

Sec. 356. Kinds of iron. Iron is distinguished into three general 
kinds : Cast iron, wrought iron, and steel. Cast iron contains a pro- 



167 

portion of carbon, and is of a brittle granulated structure. By melt- 
ing iron and stirring it while in fusion, part of the carbon is burned 
oat. Then by hammering or rolling it becomes almost pure, fibrous 
and tough ; and is then called wrought iron. After it is brought to 
the state of wrought iron, it is converted into steel by heating it in 
a confined place in contact with charcoal, with which it combines. 
It will then become hard on heating and plunging into cold water, 
and its hardness will be proportioned to the degree of heat and cold ; 
for its hardness depends on the suddenness and extent of the dimi- 
nution of temperature. 

Sec. ^ .357. Application of the kinds of iron. When great strength 
is required, or jarring and striking motion, wrought iron is prefer- 
able to all known materials. Rusting of iron, by its ready union 
with oxygen and carbonic acid, requires that it should be painted, 
or defended in some other manner. When nothing but hardness is 
required, cast iron is best. But when great hardness, joined to con- 
siderable strength is required, steel is preferable. If very great 
strength is required, bars of steel should be welded upon bars of 
tough iron, as in making sleigh-shoes. 

The smith should receive particular directions, respecting the ap- 
plication of his work, from the engineer. For example: if bolts 
are to resist great force, by pulling lengthwise of them, (as when 
holding up the string-pieces of a bridge, which is sustained by upper 
frame-work) care must be taken to have the heads large and to 
consist of a part of the bolt itself, so constructed that the heads shall 
be continuations of the bolts. The whole must be the toughest and 
softest of iron. Bolts for holding timbers from sliding or moving 
out of place may be made of iron of less tenacity, unless so situated 
as to be subject to jarring strokes, as in all moveable carriages. 



168 



Inorganic Materials for Construction,* 

With their specific gravity (see sec. 135) and weight in pounds, 
per cubic foot. It is well known that their solidity is indicated by 
their weight. 



Names. 


Speci- 
fic 
Grav. 


Weight 

in 
pounds. 


Names. 


Speci- 
fic 
Grav. 


Weiglit 

in 
pounds. 


Atmospheric Air, 
Basalt, 


.0012 
3.000 


.075 

187.50 


Limestone, compact. 
Limestone, granular. 


2.598 
2.653 


162.37 
165.81 


Brick, common, 


2.000 


125.00 


Lime, quick, stone, 


.843 


52.68 


Brick, red. 


2.168 


135.50 


Marble, Parian, 


2.837 


177.31 


Chalk, 


2.657 


166.06 


Mar], common. 


1.600 


100.00 


Charcoal, birch, 


.542 


33.87 


Moitar,paste3s-21 


1.588 


99.25 


Charcoal, oak, 


.332 


20.75 


Sand, pit, fine. 


1.523 


95.18 


Charcoal, pine, 


.280 


17.50 


Sand, river. 


1.886 


117.87 


Clay, marley. 
Coal, 


1.919 
1.290 


119.93 

80.62 


Slate, argillite, 
Wacke, gray, 


2.781 
2.614 


173.81 
163.37 


Earth, common, 


1.984 


124.00 


Wacke, grindstone. 


2.143 


133.93 


Granite, common. 


2.664 


166.50 


Sienite, 


2.621 


163.81 


Gravel, common, 


1.749 


109.32 


Tufa, calcareous, 


1.217 


76.06 


Gypsum, common. 


2.286 142.87 | 


Water, rain. 


l.OOOJ 60,N.T.t 



Organic Materials for Construction, 
With their specific gravity and weight in pounds, per cubic foot. 





Speci- 


Weight 




Speci- 


Weight 


Names. 


fic 


in 


Names. 


fic 


in 




Grav. 


pounds. 
34.68 




Grav. 


pounds. 


Alder, dry. 


.555 


Oak, live, 


1.216 


76.03 


Almond- tree, 


1.102 


68.87 


Oak, white. 


.908 


56.75 


Apple-tree, 


.793 


49.56 


Oak, red, 


.752 


47.00 


Ash, dry. 


.845 


52.81 


Pear-tree, dry, 


.708 


44.25 


Beech, partly dry. 


.854 


53.37 


Pine, pitch, dry, 


.936 


58.50 


Birch, dry, 


.720 


45.00 


Pine, white, dry. 


.460 


28.75 


Box, dry. 


1.030 


64.37 


Plane-tree, button-wood 


.648 


40.50 


Cedar, 


.753 


47.06 


Plum-tree, 


.785 


49.06 


Cherry-tree, dry, 


.672 


42.00 


Poplar, black, dry, 


.421 


26.31 


Chestnut, dry, 


.606 


37.95 


Poplar, Lombardy, dry. 


.374 


24.37 


Chestnut, horse, dry. 


.596 


37.28 


Quince-tree, 


.705 


44.06 


Ebony, 


1.331 


83.00 


Sassafras, 


.482 


30.12 


Elm, dry. 


.588 


36.75 


Sycamore, 


,645 


40.31 


Fir, black spruce. 


.512 


32.00 


Tulip-tree, white-wood, 


.477 


29.81 


Hickory, 


.929 


58.06 


Vine, grape, 


1.237 


77.31 


Hornbeam, 


.760 


47.50 


Walnut, black. 


.920 


57.50 


Lignumvitse, 


1.333 


83.31 


Walnut, butternut, dry, 


.616 


38.50 


Mahogany, 


.852 


53.30 


Willow, green. 


.619 


38.68 


Maple, dry. 


.755 


47.18 


Yew, 


.788 


48.62 



* These items of mineral and vegetable materials were selected from Treadgold's tables. 
t 62.5 lb. English, per cubic foot. 



169 



TABLES, WROUGHT EXAMPLES, &c. 

Segment of the Tahle of Logarithms of Numhers ; to be used ac- 
cording to Hutton in calculating heights of mountains, &c., by the 
barometer. 



No. 



2 



4 I 5 



6 



9 



255 

256 
257 

258 
259 



406540 
408240 
409933 
411620 
413300 



406710 

408410 
410102 

411788 
413467 



406881 
408579 
410271 
411956 
413635 



407051 
408749 
410440 
412124 
413802 



407221 
408918 
410608 
412292 
413970 



407391 
409087 
410777 
412460 
414137 



407561 
409257 
410946 
412628 
414305 



407731 
409426 
411114 



407900 
409595 
411283 



412796412964 
414472414639 



408070 
409764 
411451 
413132 
414806 



260 
261 
262 
263 
264 
265 
266 
267 
268 
269 



414973 
416640 
413301 
419956 
421604 
423246 
424882 
426511 
428135 
429752 



415140 

416807 
418467 
420121 
421768 
423410 
425045 
426674 
428297 
429914 



415307 
416973 
418633 
420286 
421933 
423573 
425208 



415474 
417139 

418798 
420451 
422097 
423737 
425371 



426836 426999 



428459 
430075 



428621 
430236 



415641 

417306 
418964 
420616 
422261 
423901 
425534 
427161 
428782 
430398 



415808 
417472 
419129 
420781 
422426 
424064 
425697 
427324 
428944 
430559 



415974 
417638 
419295 
420945 
422590 
424228 
425860 
427486 
429106 
430720 



416141 

417804 
419460 
421110 
422754 
424392 
426023 
427648 
429268 
430881 



416308 
417970 
419625 
421275 
422918 
424555 
426186 
427811 
429429 
431042 



416474 
418135 

419791 
421439 
423082 
424718 
426349 
427973 
429591 
431203 



270 
271 
272 
273 
274 
275 
276 
277 
278 
279 



431364 
432969 
434569 
436163 
437751 
439333 
440909 
442480 
444045 
445604 



431525 
433129 
434728 
436322 
437909 
439491 
441066 
442636 
444201 
445760 



431685 
433290 
4348S8 
436481 
438067 
439648 
441224 
442793 
444357 
445915 



431846 
433450 
435048 
436640 
438226 
439806 
441381 
442950 
444513 
446071 



1432007 
'433610 
435207 
436798 
438384 
439964 
441538 
443106 
444669 
446226 



432167 
433770 
435366 
436957 



432328 
433930 
435526 
437116 



4385421438700 



440122 
441695 
443263 
444825 
446382 



440279 
441852 
443419 
444981 
446537 



432488 
434090 
435685 
437275 
438859 
440437 
442009 
443576 
445137 
446692 



432649 
434249 
435844 
437433 
439017 
440594 
442166 
443732 
445293 



432809 
434409 
436003 
437592 
439175 
440752 
442323 
443888 
445448 
447003 



280 
281 
282 
283 

284 
285 
286 
287 
288 
289 



447158 

448706 
450249 
451786 
453318 

454845 
456366 
457882 
459392 
460898 



447313 

448861 
450403 
451940 
453471 
454997 
456518 
458033 
459543 
461048 



447468 
449015 
450557 
452093 
453624 
455149 
456670 
458184 
459694 
461198 



447623 
449170 
450711 
452247 
453777 
455302 
456821 
458336 
459845 
461348 



447778 
449324 
450865 
452400 
453930' 
455454 
456973 
458487 
459995 
461498 



447933 

449478 
451018 
452553 
454082 
455606 
457125 
458638 
460146 
461649 



448088 
449633 
451172 
452706 
454235 
455758 
457276 
458789 
460296 
461799 



448242 
449787 
451326 
452859 
454387 
455910 
457428 
458940 
460447 
461948' 



448397 
449941 
451479 
453012 
454540 
456062 
457579 
459091 
460597 
462098 



448552 
450095 
451633 
453165 
454692 
456214 
457730 
459242 
460747 
462248 



290 
291 
292 
293 
294 
295 
296 
297 
298 
299 



462398 
463893 
465383 
466868 
468347 
469822 
471292 
472756 
474216 
475671 



462548 
464042 
465532 
467016 
468495 
469969 
471438 
472903 
474362 
475816 



462697 
464191 
465680 
467164 
468643 
470116 
471585 
473049 
474508 
475962 



462847 
464340 
465829 
467312 
468790 
470263 
471732 
473195 
474653 
476107 



462997 
464489 
465977 
467460 
468938 
470410 
471878 
473341 
474799 
476252 

22 



463146 
464639 
466126 
467608 
469085 
470557 
472025 
473487 
474944 
476397 



463296 
464787 
466274 
467756 
469234 
470704 
472171 
473633 
475090 
476542 



463445 
464936 
466423 
467904 
469380 
470851 
472317 
473779 
475235 
476687 



463594 
465085 
466571 
468052 
469527 
470998 
472464 
473925 
475381 
476832 



463744 
465234 
466719 
468200 
469675 
471145 
472610 
474070 
475526 
476976 



170 



300 
301 

302 
303 
304 
305 
306 
307 
308 
309 


477121 
478566 
480007 
481443 
482874 
484300 
485721 
487138 
48855] 
489958 


477266 
478711 
480151 
481586 
483016 
484442 
485863 
487280 
488692 
490099 


477411 

478855 
480294 
481729 
483159 
484584 
486005 
487421 
488833 
490239 


477555 
478999 
480438 
481872 
483302 
484727 
486147 
487563 
488973 
490380 


477700 
479143 
480582 
482016 
483445 
484869 
486289 
487704 
489114 
490520 


477844 
479287 
480725 
482159 
483587 
485011 
486430 
487845 
489255 
490661 


477989 
479431 
480869 
482302 
483730 
485153 
486572 
487986 
489396 
490801 


478133 

479575 
481012 
482445 
483872 
485295 
486714 
488127 
489537 
490941 


478278 
479719 
481156 
482588 
484015 
485437 
486855 
488269 
489677 
491081 


478422 
479863 
481299 
482731 
484157 
485579 
486997 
488410 
489818 
491222 


310 


491362 


491502 


491642 


491782 


491922 


4920621492201 


492341 


492481 


492621 


1 1 1 1 2 


3!4|5|6|7!8|9 



Refraction. 

Bowditch says, that the refraction of terrestial objects may be 
found thus, very nearly correct, when the air is clear. 

Calculate the distance of the object ; and find the angle subtended 
by said distance at the centre of the earth — one fourteenth of said 
angle is the angle of refraction. 

Refraction of Heavenly Bodies in Altitude. 



App. 

Alt. 


Ref. 


App. 

Alt. 

D.M. 


Ref. 


App. 
Alt. 


Ref. 

M. S. 
13.33 


App. 
Alt. 


Ref. 
M. S. 


App. 

Alt. 


Ref. 


D.M. 


M. S. 


M. S. 


D.M. 


D.M. 


D.M. 


M. S. 


0. 


33. 


1.40 


20.18 


3.20 


6.30 


7.52 


9.50 


5.20 


0. 5 


32.11 


1,45 


19.51 


3.25 


13.19 


6.40 


7.41 


10. 


5.15 


0.10 


31.22 


1.50 


19.25 


3.30 


13. 5 


6.50 


7.31 


10.15 


5. 8 


0.15 30.36 


1.55 


18.59 


3.40 


12.39 


7. 


7.21 


10.30 


5. 


0.20 


29.50 


2. 


18.35 


3.50 


12.14 


7.10 


7.12 


10.45 


4.54 


0.25 


29. 6 


2. 5 


18.11 


4. 


11.50 


7.20 


7. 3 


11. 


4.47 


0.30 


28.23 


2.10J17.48 


4.10 


11.28 


7.30 


6.54 


11.15 


4.41 


0.35 


27.41 


2.15 


17.26 


4.20 


11. 7 


7.40 


6.46 


11.30 


4.35 


0.40 


27. 


2.20 


17. 4 


4.30 


10.47 


7.50 


6.38 


11.45 


4.29 


0.45 
0.50 


26.20 


2.25 
2.30 


16.44 
16.23 


4.40 


10.28 
10.10 


8. 


6.30 
6.22 


12. 
12.20 


4.23 


25.42 


4.50 


8.10 


4.16 


0.55 


25. 5 


2.35 


16. 4 


5. 


9.53 


8.20 


6.15 


12.40 


4. 9 


1. 


24.29 


2.40 


15.45 


5.10 


9.37 


8.30 


6. 8 


13. 


4. 3 


1. 5 


23.54 


2.45 


15.27 


5.20 


9.21 


8.40 


6. 1 


13.20 


3.57 


1.10 


23.20 


2.50 


15. 9 


5.30 


9. 7 


8.50 


5.55 


13.40 


3.51 


1.15 


22.47 


2.55 


14.52 


5,40 


8.53 


9. 


5.49 


14. 


3.46 


1.20 


22.15 


3. 


14.35 


5.50 


8.39 


9.10 


5.43 


14.20 


3.40 


1.25 


21.44 


3. 5 


14.19 


6. 


8.27 


9.20 


5.37 


14.40 


3.35 


1.30 


21.15 


3.10 


14. 3 


6.10 


8.15 


9.30 


5.31 


15. 


3.30 


1.35 


20.46 


3.15 


13.48 


6.20 


8. 3 


9.40 


5.26 


15.30 


3.23 



171 



App. 

Alt. 


Ref. 


App. 
Alt. 


Ref. 


App. 
Alt. 


Ref. 


App. 
Alt. 


Ref. 


D.M. 


M. S. 


D. 


M. S. 


D. 


M.S. 


D. 


M. S. 


16. 


3.17 


30 


1.38 


50 


0.48 


70 


0.21 


16.30 


3.11 


31 


1.35 


51 


0.46 


71 


0.20 


17, 


3. 5 


32 


1.31 


52 


0.45 


72 


0.19 


17.30 


2.59 


33 


1.28 


53 


0.43 


73 


0.17 


18. 


2.54 


34 


1.24 


54 


0.41 


74 


0.16 


18.30 


2.49 


35 


1.21 


55 


0.40 


75 


0.15 


19. 


2.44 


36 


1.18 


56 


0.38 


76 


0.14 


19.30 


2.40 


37 


1.16 


57 


0.37 


77 


0.13 


20. 


2.36 


38 


1.13 


58 


0.36 


78 


0.12 


20.30 
21. 


2.32 

2.28 


39 
40 


1.10 


59 


0.34 
0.33 


79 


0.11 


1. 8 


60 


80 


0.10 


21.30 


2.24 


41 


1. 5 


61 


0.32 


81 


0. 9 


22. 


2.20 


42 


1. 3 


62 


0.30 


82 


0. 8 


23. 


2.14 


43 


1. 1 


63 


0.29 


83 


0. 7 


24. 


2. 7 


44 


0.59 


64 


0.28 


84 


0. 6 


25. 


2. 2 


45 


0.57 


65 


0.27 


85 


0. 5 


26. 


1.56 


46 


0.55 


66 


0.25 


86 


0. 4 


27. 


1.51 


47 


0.53 


67 


0.24 


87 


0. 3 


28. 


1.47 


48 


0.51 


68 


0.23 


88 


0. 2 


29. 


1.43] 49 


0.50 


'69 


0.22 


89 


0. 1 



Extensive factor for finding the circumference of a circle, when the 
diameter is given, ly multiplying it into the factor. 

3.14159265358979323846264338327950288419716939937510 
582097494459230781640628620899862803482534211706798214 
80865132723066470938446 + 



Long Measure. 

7.92 inches make one link. 
100 links (66 feet) one chain. 
80 chains make one mile. 

Superficial Measure. 

10 square chains make one acre. 

640 square acres make one square mile. 



172 



Cubic Measwe computed in Weight. 

28.8 cubic inches (one pint) of pure water weigh one pound. 

A cubic foot of pure water weighs 60 pounds. 

2000 pounds (2000 pints) make one ton. 

Note. This is according to the revised statutes of New- York. 
In England, and many of the States, a cubic foot of water weighs 
62.5 m. A ton, 2240 ife. 

Five gallons per day is allowed in Paris (France) for each indi- 
vidual, at average, by the waterworks companies. In^ Troy 3-|- 
gallons per day. Partly supplied by cisterns and wells. 

Cord Wood. 

Multiply length, breadth, and thickness together, and allow 128 
cubic feet for a cord. If the measure is in feet and inches, multiply 
the length 8 f. 4 in. breadth 4 f. 7 in. height 3 f. 9 in." This is 
called duo-decimal rule. 

8 4 38 2 4 128)143 2 9(1 

4 7 3 9 128 



38 


2 4 


3 


9 


114 


7 


28 


7 9 


143 


2 9 


Bushel Measures. 



33 4 114 7 15 2 9 

4 10 4 



38 2 4 143 2 9 1 cord, 15 f. 3 in. 



A bushel of charcoal, if birch, weighs 45.75 ife. — if oak, 28 ife. 
To find the number of bushels in a load of charcoal, find the con- 
tents in cubic inches, as directed in sections 221 and 222. Divide 
the contents by 2339 (the cubic inches in a bushel) the quotient will 
be the measure in bushels. But an allowance of 14 per cent, for 
shaking on a wagon is allowed, if it has been moved a considerable 
distance. 114 bushels, on being drawn from Sandlake, was shaken 
down to 100 bushels. Apples will shake down about 4 per cent. — 
also ears of corn and potatoes. 

Charcoal ought always to be sold by weight. Let one bushel 
be weighed for a standard ; then weigh the load on the hay -scales ; 
then re-weigh after the quantity sold is taken out. 



173 

Comparing Rail-Road Curves, from sec. 217. 

When radius 2000 feet, arc 500 feet. 
When radius 1500 feet, arc 666.66 feet. 
When radius 800 feet, arc 1250 feet. 
All describe areas of 500.000 feet each. 

2000 rad. 
500 arc. 



2)1,000,000 double area. 
500.000 area, 
rad. 1500)1,000,000 double area. 



666.66 arc with radius of 1500 feet. 
rad. 8.00)1,000,000 double area. 



1250 arc with radius of 800 feet. 

W. G. L. 

From section 287. 

Formula for pipes when the diameter is sought. 

The quantity of water discharged per second is 3 cubic feet. 
The length of the pipe 4000 feet. 
Head, or descent 10 feet. 



3 cubic feet. 


38.116 


3 


38.116 


9 


228696 


4000 


3S116 




38116 


36000 


304928 




114348 


38.1162 = 


14528.29456; 



174 

14528.29)36000.000000(2.47792 
2905658 



6943420 
5811316 

11321040 
10169803 

11512370 
10159803 

13425670 
13075461 

3502090 
2905658 



Extracting the root according to Hutton's method for the higher 
powers. See sec. 30. 

Assuming 1.19 for the root. 

1.19 is raised to the fifth power 2.386. 



2.386 

2 


2.47792 
2 


; 1.19 
: 1.2 : 




4.772 
2.47792 


4.95584 

2.386 




7.24992 : 

Second statement : 

1.25=2.48832 
2 


7.34186 : ; 

: 2.47792 
2 


: 1.2 


4.97664 
2.47792 


4.95584 

2.48832 




5.45456 


: 7.44416 : 


1.1983. 
J.O. 



175 



Calculation, from section 288. 

Formula for open canals when the velocity and quantity are re- 
quired. 

Head 10 feet. 
Length 30 feet. 
Area 4.8. 
Perimeter 6.8. 

10x4.8= 48. 
30x6.8=204.)48.00009(.23529 

9582 



47058 
188232 
117645 
211761 

2254.54878 
.0111 

2254.55988(47.482 
16 0.109 * 



87)654 47.373 velocity in feet per sec. 

609 4.8 transverse area. 



944)4555 378984 

3776 189492 



9488)77998 227.3904 cubic feet X 60 ft. = 
75884 13643.4240 ft. per sec. 



94862)211480 E. N. H. 

189724 

* In section 28S, this fonniUa is wrongly directed to be added. 



176 

Calculation of the height of the Atmosphere. 

From sections 292 and the sixth subdivision. 

Hours. Min. 

24 : 360° :: 70 

60 70 



Min. 1440 1440)25200(17.5=17° 30' 

1440 



10800 
10080 

7200 
7200 



Sine of Sem. Diam. Sine of 
81° 15' Miles. 90° 

.98836 : 4000 : : 1.00000 



4000 

Miles. 

.98836)4000.00000(4047.1 from earth's 

395344 centre to the top of 

atmosphere. 



465600 
395344 

702560 
691852 

107080 
98836 



4047.1 

4000 earth's radius. 



47.1 height of atmosphere. Generally estimated at 45 miles. 

A. B. C. 



177 



WOOD.CUT FIGURES; 

Explaining and familiarizing some of the subjects treated of in the 
preceding sections. 

Lines are right or curved. A right line is 

the shortest measure between two points ; as 

in the figure. A curved line is always a line, which, if sufficiently 

>*— >. y-^ ^— ^ continued, will return into itself, and form 

^■^ ^"^"^ ^ a circle ; as each of the curves in the 

figure. 



Circle, is a figure bounded by a continued line, 
every where equi-distant from a point in the cen- 
tre, c sector, bounded by two radii and an arc ; 
d segment, bounded by a chord line and an arc ; 
a the centre. 

Ellipse, made by moving one pin's point around 
two focal ones ; and is kept in the periphery by 
sliding around in the loop of a thread. 






Square, is a figure of four equal sides meeting at right 
angles. 



Rectangle, (or parallelogram) a long square 
with opposite sides only equal. The line con- 
necting opposite corners is a diagonal. 

Rliomh, a figure with four opposite equal 
\ sides, not meeting at right angles. 



Superficies, having length and breadth without 
thickness. 



Solid, having length, breadth, and thickness. 
23 



178 



Right angle, a BqaaTecotneT. Sec. 31. 



Obtuse angle, opens -widet than a right angle. Sec. 



V 



31. 



jf Acute angle, does not open as wide as a square. Sec. 
51. 



A right-angled triangle contains one right angle. Sec. 
32. 




Ohtuse-angled triangle has one obtuse an- 
gle. Sec. 32. 

Acute-angled triangle has all three angles acute. 
If it has two equal sides, it should be called also Isos- 
celes triangle. If the three sides are equal, it is also 
called Equilateral. Sec. 32. 



a c is the sine of the angle e. Sec. 33, article 3. 




%/\ V The figured line is a line of chords. Sec. 33, 
"^rV y article 4. 



Illustration of the principle that a tri- 

angle contains 180 degrees. Semi-circle 

l|~~ g i is the measure of 180° — angle a in it 

equals angle a in the triangle — e equals 

e — c equals c. Sec. 33, article 5. 



17» 




Geometrical trigonometry, is illustrated by plotting 
this figure ; having the side given, whicli is drawn from 
the angle at 80 to the angle at 70, and drawing the lines 
up to g, through the marked points on the arcs d and /, 
until they cross at g. See sec. 34. 



A 



/ B \ 




Frustrum of a cone or pyramid, found 
by calculating the pyramid as if topped 
out ; and then calculating and subtracting 
the added point. See sec. 53, article 6. 

This figure also illustrates the 7th arti. 
cle, Guaging, under the 53d section. 




Field surve3ring with 
a cross, where a farmer 
is desirous to know the 
contents of a field for 
planting seed, to pay for 
mowing by the acre, &c. 
See sec. 60 and 61. 



ISO 

Map of a farm referred to from sections 74, 75, 78, and then in 
all the sections to sec. 89. And again from section 106 to 109» 
where heights and distances and division of land are explained. ■^ 




181 

Plot of a survey referred to from sections 91 to 95 ; wherein the 
method of reducing and raising the scale, plotting, triangular cuttings 
and castings are explained. 




182 

Plot of a survey, wherein the area is cast up by reducing the 
whole'survey of a single triangle. This is referred to and explained 
from sections 96 to 99. 




Contraction of the vein, from sec. 315. 

One third of the area is generally deducted for the contraction 
of the vein. This expressed decimally is 0.666 — this agrees nearly 
with Bossut's experiments. Eytelwein adopts 0.640. It has been 
demonstrated by experiment, that a conical tube, whose length is 
about 0.88, the diameter of its base, if adjusted to the aperture, will 
reduce the contraction of the vein to about one sixth — that is, the 



183 

area of the contracted vein will be but one sixth less than that of the 
area of the aperture. Also, that a gate-hole through a four-inch 
plank, cut a little convergingly, will add much to the efflux of a co- 
lumn of water ; by lessening the contraction of the vein. 



Plot of a survey calculated by the trapezoidal metTioar-Tererred 
to from sections 100 to 104. 




184 

Harbor survey ; being a section of Hudson river, above Troy 
Referred to from sections 120 to 126. 

■■■■-'■■ m 

'r'W 



■■ ■■ Mi 



-■/.mA 1 



-m 



||\ 9o:m 




■■■' .•■■"■■•■ # 
/■ ,^3* ■■II \ 



■Mi.. 



■■■mm 
..-•■■ ..-ill 

" ■■il .ll 
■■■/■■¥ 



::t6S 



185 

For sec. 198 to 200, on Rail-Road Curves. 

Two propositions, referred to in sec. 199, are here given at full 
lenglli. 

1. Two chord lines meeting in the periphery of a circle, form an 
angle, which is measured by an arc, half as long as the arc requir- 
ed to connect the ends of two radii, which meet the ends of the chord 
Imes. As the arc A H C is twice as long as an arc required to 
measure the angle ADC. 

2. Two sub-chord lines meeting at any point in the periphery of 
a circle, the point of meeting will continue to form the same angle, 
if moved to any other point of the periphery, on the same side of 
the general chord line. As the angle 130° at B, is 130° at D. 

Description of tlie figure referred to sec, 198 to 200. 

Scale 200 feet per inch. Radius E C 300 feet. General chord 
hne A C 465 feet. Sub-chord line A B is a traverse line 360 feet 
long, taken in the field. Sub-chord line B C is a traverse line 132 



^^^^ 




2325tf 



■■% 










4 




■. ^r5tf^• 






H^'" 






IOO°E 





^c of 260 

24 



186 

feet long, taken in the field. Angle B at the meeting of the tra- 
verse lines, is 130°; as found by considering the compass directions 
of each. Moved to D, it continues to be 130°, according to the se- 
cond proposition above. This angle doubled makes 260°, and is 
measured by the arc A H C, according to the first proposition above. 
This subtracted from 360°, gives the angle 100° at E. The diag- 
onal line E D halves the angles at E and D. The general chord 
line A C, being halved by said diagonal line, give horizontal legs 
to four right-angled triangles; to wit, K L M N, each being 232|^ 
feet. 

Ahwe description extended to the second figure ; which is referred to 
sec. 201 to 204, 

Tan. the tangent fine, from which the line a J is deflexed 10°, a 
number equal to half the angle (20°) of the isosceles triangle at the 
centre of the circle. The next line c d is deflexed from the chord 
line a c 20°, a number equal to the angle at the apex of the isos- 
celes triangle at the centre of the circle. All the remaining de- 
flexions are 20° also ; as they are deflexed from the last preceding 
chord line, which is double the deflexion from the tangent line. 




IQO' 



187 



Rail-Road Curve. Illustration of sec. 207. 

The dotted lines are used in sec. 208. This example comprises 
more than half a circle ; of course the line P W is shorter tlian 
P V. In practice no curve ever includes half a semi-circle. 




Sliding Rule. 

The sliding rule has four Hnes — two stationary on the wooden 
part, two sliding ones on the brass slip. The upper one on the wood 
is marked A — the upper one on the brass is marked B — the lower 
one on the brass is marked C — the lower one on the wood is mark- 
ed D, and called girt line. 

Measuring boards. 

1. Take the width of the board in inches. 

2. Find the number agreeing with the number of inches on line 

A, and shde figure 12 to it on B. 

3. Read off the square feet by measuring the length of the board 
in feet and inches, and finding the number agreeing with it on line 

B, and against it on A, read the square measure of the board in feet. 

Measuring timber, square or round. 
1. Take the length of the timber in feet and inches. 



188 

2. Find the number agreeing with the number of feet and inches 
on the hne C, and sHde it to the figure 12 on D. 

3. Read-off the cubic feet hj finding a quarter of the girt in inches 
and finding a number agreeing with the inches of the quarter girt on 
the line D, and against it on C, read the cubic contents of the timber 
in feet. 

For sections from 211 to 214. Illustration of the Calculation of 

Ordinates. 

B C the base (100 feet) A B and A C the two equal side& (300 
feet each) of an isosceles triangle. A, the angle at the apex of 
the isosceles triangle (20°) being double the angle formed by the 
deflexion of the chord line B C from the tangent line B a (10°.) 
The angle at A is at the centre of the circle, of which B 10 C is an 
arc. E D is a middle portion of the horizontal diameter of the 
circle ; which, in its whole length, is double the radii A B and 
A C = 600 feet. E D being equal to the chord line of the given 
arc (100 feet) it has 50 feet on each side of the centre A. There- 
fore the distance from D to the end of the diameter in the direction 
of E, is 350 feet, and the remainder in the opposite direction is 250 
feet. By a known principle in mathematics, if 350 is multiplied by 
250, and the square root of the product extracted, it will give the 
length of the ordinate D C. In the same manner the line E B is 
found. Shorten the side in the direction of E, 5 feet to c, leaving 
that line 345 feet, and making the other 255, and intermultiply them 
and extract the square root as before, you obtain the ordinate c n. 
Subtract the standing ordinate D C from the ordinate c 7i, and the 
remainder will be that part of the ordinate which is above the chord 
line B C. Proceed in the same manner with v s, and all other 
m'easures on the diameter line E D, moving 5 feet at a time along 
said line. In this manner all the offset lines (called ordinates) above 
the said chord hne, are obtained, as far as the middle ordinate A W. 
Then by ktverting their order, all between W and B may be set 
down. In this manner the table under section 210 was made. 



189 




190 



For sections 224, 225 and 226. 



In taking the cross areas of excavations and enibanhnents, it is 
generally preferable, when very irregular, to suppose the base and 
surface level, and of course, parallel, however uneven they may be 
in reality. Then add their calculated lengths, halve their sum, and 
multiply that by the distance between their levels. This imaginary 
trapezoidal result is then to be reduced to the truth, by casting and 
subtracting the vacant places. 

Examvle, Fig 1. A B is a side hill. Cast the trapezoid B C E G 
in the usual way. Cast the triangle A B C in the usual way, when 
a triangle is made between two parallel lines, A D and B L. Ur 
the whole trapezoid, A D E G, may be cast, and the triangle AD B 
be deducted. 0^ In all cases the distance between parallel lines, 
and between an apex and a base, the difference of levels is the dis- 

Examvle Fig. 2. Z T O m N H is the uneven surface of a piece 
of required'excavation. Add Z X and R W, halve their sum and 
multiply that by the distance between the ascertained level ot the 
base R W, and the level of the highest point to be excavated, Z A ; 
then cast and subtract the vacant spaces. First, cast the trapezoid 
Z X T S, in the usual way. Second, cast triangle HNS, whose 
base is calculated from the central hench (fix^ed stake) and whose 
perpendicular is the difference in level between the apex at H and 
the base N. Third, consider the figure T N O t., as a trapezoid, 
and calculate it as such; for though the side O uis not parallel io 
T N, it is a case which may be averaged by the levelled hne O M 
being made to be intermediate in height between the true levels of 
O and u. Fourth, add the areas of these two trapezoids and the 
triangle, and subtract their sum from the factitious area of the whole 
assumed trapezoid. 

Remark. This transverse area may also be cast, directly, by 
casting the trapezoid R W K H-then the trapezoid K H V r- 
then the triangle V O Z-then, last, make an average triangle u r N. 
But the point would require a special measure, and both ends of the 
base line u r, would require more time and become more comph- 
cated, than by adopting the deduction method. In all cases the sur- 



191 

face is most accessible ; consequently may generally be easiest 
measured. Besides, the engineer can always construct his general 
ideal trapezoid in perfection, and fix some points of it to his benches. 





Power to overcome friction in a flouring mill, from sec. 315, con- 
tinued. 

Suspend weights on a water-wheel at its periphery on a horizon- 
tal level with its axis, cog-wheel, or other vertical wheel, until the 
whole gearing, stone, &c., start. Then calculate the wheel and 
axil power, so as to compare the advantage at the point of the appli- 
cation of the weight, with the point (or« average point) where the 
water acts on the wheel, whether overshot, undershot, or horizontal. 



INDEX 



Pages. 
131, 132 
85 
76 
10 to 25 
56, 57 
134 
80 
90 
172 to 176 
117 to 125 
133 
13 
156, and on. 
136, 137 
172 
Convexity of the earth, 111 

Contraction of the vein, 143, 182 
Curves, 104 to 112, 185 



Accelerating forces, 
Architecture, 
Areometer of Baum, 
Arithmetic, 
Ascent and descent, 
Atmospheric height, 
Atmospheric pressure. 
Barometer, 

Calculations of examples 
Canals, 

— formula, 

Characters, 
Classes of strata 
Clouds, 
Coal measure, 



running, 
comparing. 



Decimals 

Density of materials for con 

struction, 
Displacement of water, 
Dynamics, 
Ellipse, 
Embankments, 
Evans' directions. 
Excavations, 
Falling bodies, 
Field book, 
Flumes, 

Friction of carriages, 
— of mills. 



100 

111 

14 

168 

117 

74 

111 

113, 190 

100, 115 

113,190 

75 



58, 69, 97, 98, 115 
142 



Funicular power, 
Genicular power. 
Geology, alphabet, 
series or 



129 
191 

83 

83 

152 

158 

82 

83 

83 

45. 179 

72', 184 

55, 56 

78 



Gonatous power, 

funicular, 

genicular, 

Guaging, 

Harbor survey. 

Heights and distances. 

Hydrodynamics, 

Hydrostatics, 77 

Iron materials, 167 

Jupiter's moons for longitude, 93 

Kendall's tangent scale, 50 

Land surveying, 47 to 73 

Latitude, 92 

Locks, 121, 122 

filling and emptying, 124 

Longitude, 93, 94 

Materials for construction, 148 to 168 
Mechanical powers, 84 

Mensuration, 42, 177 



Pages. 
Mills, 143, 144, 145, 146, 147 

Millstones, 147, 148 

trituration, ' 147 

Moon's distance, under sec. 44, 

note, 34 

Moving bodies on water, 117, 118 
Natural Sines, 



Notation, 

Offsets, 

Ordinates, 

Pipes, 

Poestenkill mills, 

Pump, 

Bail-road, 

survey, 

extemporaneous 

preliminary, 

definite. 



Random lines, 
Retarding forces, 
Roads in general, 

surveying. 

Roots, square, 

cube, 

high powers, 



Rule of three, 
Sargeant's directions. 
Scale, reducing, 

raising. 

Shades for roads. 
Sines, natural, 

abridged, 

Single triangle. 
Sliding Rule, 
Specific gravity, 



29 
11 

54, 180 

108,188 

132,133 

143 

81 

89 to 116 

89, 92, 95 

, 89 

95 

98 

55, 180 

131, 132 

125 

68 

19 

21 

24 

17 

95 to 100 

60 

62 

127 

19 

viii and 31 

62,182 

187 

76 



of materials for 



construction, 168 

Starting point in staking a curve, 105 
Statics, 74 

Supply of water for a canal, 123 

for a mill, 138 

Surveying, 47, 89, 92, 95, 100, 180 



Table of logarithms, 

of refraction, 

Timber materials. 

Topography, 

Trapezoid, 

Tiigonometry, 

Useful rocks, , 

Velocity, 

Water as an agent. 

its powers, 

under pressure. 

works. 



Weirs, or water-pitches, 

table for it, 

Wood-cut figures, 



169 

170 

165 

148 

64, 183 

26, 32, 178 

162 

75, 138 to 140 

75 

137 

80 

131 

145, 147 

141 

177 to 191 



*X 



6"^ 



rV V - 



■V- 






o 0^ 










.Oo 









'^^ '^'%\- 







.A- 



S'''^. 















Deacidified using the Bookkeeper process. 
Neutralizing agent: Magnesium Oxide 
Treatment Date: Jan. 2004 

PreservationTechnologies 

A WORLD LEADER IN PAPER PRESERVATION 

1 1 1 Thomson Park Drive 
Cranberry Township, PA 160S6 
(724) 779-21 1 1 



■ "^ ' 


'b. 






I'St. ' 


'^ 






: -^ '■ 




•>' 








o 0' 






,4 


^^ 






A' 



















.^C^ 






^ ' , X " , 



■?.. ''' 



."^ 



-^^ V^' 



-^^^^V ^^ ^'^. -.^ 



:>° .0- 



V-^:^^^^ . V 



.'^■ 






.0^ . 



.^^ 












5 -^ :i 












.A^\ ,>^^''« ' 



%; 



■j 



.is- 






.0 -o 






■^^. r^^ 



O 0' 
o5 -nt-. ',W 



^0^ 



-^''' ^"^ 




^^ v- 



?. oH 

X-" a.^^ 

\^^^ 



\' X 



*■<(• 



<^' 



"^^ v^"* 



s^^ 






^ .0- 



-/ -1^ 



-o- 









.'^V^ C 



o 



.0 



■^^' 



'OO' 









-"'J v^' 






•''<^..N<' 









■^^, , A 



